r 


REESE   LIBRARY 


__n— n— ix 


UNIVERSITY  OF  CALIFORNIA. 


I 


ROPE-DRIVING: 


A   TREATISE  ON  THE  TRANSMISSION 

OF  POWER    BY  MEANS'  OF 

FIBROUS  ROPES. 


BY 

JOHN  J.  FLATHEB,  Pn.B.,  M.M.E., 

Professor  of  Mechanical  Engineering,  Purdue  University. 


FIRST  EDITION. 

FIRST  THOUSAND. 


NEW  YORK : 

JOHN    WILEY   &    SONS. 
LONDON:   CHAPMAN  &  HALL,  LIMITED. 

1895. 


Copyright,  1895, 

BY 

JOHN  J.  FLATHER. 


ROBERT  ORUMMOND,   ELECTROTYPER  AND   PRINTER,   NEW  YORK. 


PKEFACE. 


THE  following  treatise  lias  been  prepared  to  supply  the 
existing  need  of  a  comprehensive  manual  of  practical  in- 
formation concerning  rope-driving  and  the  principles  npon 
which  the  practice  rests. 

In  its  preparation  free  use  has  been  made  of  whatever 
literature  could  be  found  relating  to  the  subject,  and  refer- 
ences for  further  investigation  are  given  in  foot-notes 
throughout  the  work. 

Most  of  the  data,  however,  have  been  collected  by  the 
writer,  who  desires  to  acknowledge  his  indebtedness  to 
those  who  have  assisted  in  the  work  by  furnishing  infor- 
mation and  drawings.  Especially  to  be  mentioned  are  Mr. 
C.  W.  Hunt,  the  well-known  authority  upon  this  subject; 
and  Mr.  Spencer  Miller,  long  identified  with  the  practice 
of  rope  transmission. 

J.  J.  FLATHER. 

LAFAYETTE,  INC.,  Oct.  1895. 


OF  THE 

UNIVERSITY 


CONTENTS. 


CHAPTER  I. 
INTRODUCTION 1 

Leather  belts  and  spur-gears — Early  use  of  ropes — Advan- 
tages of  ropes— Losses  in  various  systems  of  transmission. 

CHAPTER  II. 

MULTIPLE  ROPE  SYSTEM 10 

Rope-wells — Distribution  of  power  to  several  floors— Influ- 
ence of  arc  of  contact— Rope-splicing. 

CHAPTER  III. 

CONTINUOUS  ROPE  on  WOUND  SYSTEM 25 

Vertical  transmission  through  several  floors — Tension-car- 
riages— Rope-tightener — Contraction  of  ropes — Transmission 
at  an  angle— Side  lead— Multiple  idlers— Dynamo  driving — 
Jack-shafts — Coil- friction — Winder-pulleys — Application  of 
ropes  to  water-wheels — Rochester  plant. 

CHAPTER  IV. 

LONG  DISTANCE  TRANSMISSION 62 

Hirn's  use  of  steel  band — Wire  ropes — Use  of  shafting— 
Limit  of  length  for  shafting — Draw-rods — Efficiency  of 
power  transmissions. 

CHAPTER  V. 
FIBROUS  ROPES 75 

Rawhide  — Leather — Manilla — Cotton — Structure  of  cotton 
fibre — Strength  of  cotton  ropes— Manilla  fibre. 

CHAPTER  VI. 

MANUFACTURE  OF  ROPES 05 

Size  of  yarns — Degree  of  twist — Lambeth  rope — Lubrication 
of  ropes— Stevedore  rope — Effect  of  tar  on  ropes— Area  of 
rope — Strength  of  manilla  ropes — Factor  of  strength  and 
wear — Working  strain — Size  of  ropes. 

T 


VI  CONTENTS. 

CHAPTER  VII.  PAGE 

WEAK  OF  ROPES 103 

Influence  of  pulley  diameter — Internal  wear — External  wear 
— Harmonic  vibration — Life  of  ropes — Weight  of  ropes. 

CHAPTER  VIII. 

HORSE-POWER  TRANSMITTED  BY  ROPES Ill 

Coefficient  of  friction — Difference  in  tensions — Influence  of 
centrifugal  force — Graphic  representation — Relative  cost  of 
rope-driving. 

CHAPTER  IX. 

DEFLECTION  OF  ROPES 128 

The  catenary — Approximate  equations — Table  of  deflections 
— Method  of  laying  out  curve — Inclined  transmissions — Ten- 
sion-weight for  given  deflection. 

CHAPTER  X. 

LOSSES  IN  ROPE-DRIVING 141 

Engine- friction — Effect  of  temperature — Friction  of  shafting 
—  Power  absorbed  by  shafting. 

CHAPTER  XI. 

LOSSES  IN  ROPE-DRIVING — Concluded 159 

Resistance  due  to  bending — Wedging  in  the  grooves — Effect 
of  groove  angle — Wood  rims — Differential  driving — Creep 
of  ropes. 

CHAPTER  XII. 

CONSTRUCTION  OF  ROPE  PULLEYS  1 77 

Least  diameter  of  pulley  —  Uniformity  of  pitch — Milled 
grooves — Cast  grooves — Light  pulleys — Wood-filled  pulley 
rims — Rim  sections — Proportions  for  groove — Idlers— Hubs 
— Double  arms — Split  pulleys — Diameter  of  bolts — Sections 
of  arms — Stresses  in  pulley — Methods  of  joining  arms  and 
rim — Built-up  rope-pulleys. 


CONTENTS.  yii 

TABLES. 

PAGE 

1.  Limit  of  Length  for  Steel  Shafting 70 

II.  Limit  of  Length  for  Wrought-iron  Shafting 71 

III.  Strength  of  Cotton  Transmission  Ropes 81 

IV.  Strength  of  Manilla  Transmission  Ropes 95 

V.  Executed  Rope  Transmissions 96 

VI.  Greatest  Revolutions  per  Minute  for  given  Diameter  of 

Rope 104 

VII.  Weight  of  Ropes 110 

VIII.  Values  of  1-2  for  a  Working  Stress  Equivalent  to  200d2 

pounds 117 

IX.  Angle  embraced  by  Rope 117 

X.  Frictiou  and  Stress  Moduli 118 

XI.  Horse-power  Transmitted  by  Ropes 121 

XII.  Relative  First  Cost  of  Rope-driving 123 

XIII.  Relative  Wear  and  Cost  of  Rope  per  Horse-power  Trans- 

mitted    125 

XIV.  Deflection  of  Rope 134 

XV.  Friction  Per  Cent  under  Varying  Loads 143 

XVI.  Power-absorbed  by  Friction  in  Line-shaft 152 

XVII.  Power  absorbed  by  Friction  in  Jack  or  Head  Shafts 154 

XVIII.  Power  absorbed    by    Lead-shaft    carrying    High-speed 

Ropes 156 

XIX.  Angle  of  Grooves  for  Equal  Adhesion 169 

XX.  Values  of  #•'  and   ^V , 180 

XXI.  Least  Diameter  of  Pulley  for  given  Diameter  and  Speed 

of  Manilla  Rope 180 

XXII.  Rope-pulleys  for  General  Work 181 


OF  THE 

UNIVERSITY 


ROPE  -DRIVING 


CHAPTER  I. 

ALTHOUGH  toothed  gearing  and  belts  are  the  most  fa- 
miliar mediums  for  the  transmission  of  power  in  mills  and 
factories,  systems  of  rope  transmission  for  this  purpose 
have  been  in  use  for  many  years,  but  it  is  only  within  the 
last  ten  years  that  they  have  given  promise  of  being  gen- 
erally recognized  in  this  country  as  a  convenient  and  effi- 
cient means  of  accomplishing  the  ends  for  which  they  were 
designed.  Until  about  fifty  years  ago  it  was  generally 
thought  by  engineers  that  cotton-  and  woollen-mills,  and 
all  others  requiring  a  considerable  amount  of  power,  could 
not  be  run  effectively  without  large  and  ponderous  lines  of 
upright  and  horizontal  shafts  of  either  cast  or  wrought  iron 
and  heavy  trains  of  gear  wheels. 

When  large  leather  belts  began  to  be  introduced  as  a 
substitute  for  gears  it  was  thought  to  be  an  experiment  of 
very  doubtful  result,  if  not  altogether  impracticable;*  but 
when  high  speeds  and  lighter  shafting  were  used  in  con- 
junction with  the  wide  belts,  the  marked  success  which  at- 
tended their  general  adoption  in  America  during  the  next 
twenty  years  attracted  considerable  attention  in  England. 
It  was  not,  however,  until  about  thirty  years  ago  that  the 

*  Journal  Franklin  Inst.,  1837. 


ROPE- DRIVING. 

American  system  began  to  replace  the  old-fashioned  gear- 
ing. The  belt  system  made  very  slow  progress  in  England, 
however,  and  before  it  had  been  at  all  extensively  adopted 
a  newer  method  was  introduced  and  quickly  came  into 
prominence,  making  such  rapid  progress  as  to  almost  en- 
tirely supplant  the  old  wheels. 

The  use  of  hemp  rope  for  power  transmission  had  been 
revived  about  1860  by  the  Messrs.  Combe  &  Barbour  of 
Belfast,  who  introduced  it  successfully  into  small  mills  in 
the  north  of  Ireland.  This  was  followed  by  its  speedy 
adoption  in  the  jute-mills  of  Dundee,  and  subsequently  in 
the  cotton-mills  of  England. 

Previous  to  this,  fibrous  ropes  had  been  in  use  for  trans- 
mission purposes,  but  their  application  had  been  limited. 

At  the  Colonial  Eope  Factory  at  Great  Grimbsy,  in  Lin- 
colnshire, from  1830  to  1837,  ropes  had  been  employed  for 
taking  off  the  power  from  the  engine  and  communicating 
to  the  first  motion  shaft.  The  plan  was  very  commonly 
adopted  in  connection  with  rope-works,  where  the  driving 
rope  employed  was  known  as  the  fly-rope  for  working  the 
equalizer  from  end  to  end  of  the  ropewalk.  The  machin- 
ery in  connection  with  the  flax-spinning  mills  at  the  same 
place  was  also  driven  by  means  of  rope  gearing.* 

Ropes  were  also  used  in  this  country  many  years  before 
their  more  general  adoption  in  England  which  followed  the 
movement  introduced  by  Messrs.  Combe  &  Barbour.  In  a 
recent  communication,  Mr.  Samuel  Webber  states  that  he 
remembers  the  occasional  use  of  rope-driving  for  temporary 
purposes  ''back  in  the  forties/'  In  one  case  he  mentions, 
power  was  carried  from  a  small  engine  outside  through  a 
window  into  a  mill  to  grind  cards  before  the  wheels  and 
main  belts  were  ready. 

The  use  of  ropes,  however,  was  not  common,  and  it  has 

*  Mr.  W.  Smith,  Proc.  I.  M.  E.  1876,  p.  393. 


ROPE-DRIVING.  3 

only  been  in  recent  years  that  rope-driving  has  come  into 
prominence  as  a  factor  in  power  transmission. 

By  this  system,  according  to  one  of  its  earlier  promoters,* 
Mr.  Jas.  Durie,  large  powers  were  transmitted  "by  means 
of  round  ropes  working  on  grooved  wheels,  which  in  some 
parts  of  this  country  [England]  have  been  largely  sub- 
stituted for  toothed  gearing.  In  this  mode  of  driving,  the 
fly-wheel  of  the  engine  is  made  considerably  broader  than 
the  fly-wheel  of  an  engine  having  cogs  on  its  circumference; 
and,  instead  of  cogs,  a  number  of  parallel  grooves  for  the 
ropes  are  turned  out,  the  number  and  size  of  which  are 
regulated  by  the  power  to  be  taken  oft'  the  fly-wheel.  The 
power  which  each  of  the  ropes  will  transmit  depends  upon 
their  size  and  the  velocity  of  the  periphery  of  the  fly-wheel/' 

As  rope-driving  has,  until  recent  years,  been  a  matter 
largely  of  experiment,  the  results  which  have  been  obtained 
from  its  use  have  not  always  been  of  uniform  excellence, 
mainly,  however,  because  designers  have  failed  to  properly 
recognize  the  requirements. 

As  the  conditions  under  which  the  systems  were  installed 
have  been  so  varied,  it  is  not  surprising  to  find  many  cases 
where  the  ropes  have  been  rapidly  worn  out  and  replaced 
by  leather  lelting,  or  other  methods  of  transmission;  but 
where  rope-driving  has  been  tried  and  has  failed,  investi- 
gation will  invariably  show  the  absence  of  suitable  condi- 
tions or  a  disregard  of  correct  principles  of  design  or  con- 
struction. In  many  applications  too  great  a  strain  is  put 
upon  the  rope,  and  the  stretch  and  wear  are  rapid;  in  ad- 
dition to  this,  the  pulleys  are  often  of  unsuitable  size,  and 
the  rope  is  unnecessarily  weakened  through  the  action  of 
the  fibres  upon  one  another.  Both  of  these  causes  are  a 
constant  source  of  annoyance  in  a  rope-drive  which  has 
been  poorly  designed  or  constructed.  A  case  in  point  can 


*Proc.  I.  M.  E.  187$, 


4  ROPE-DRIVING. 

be  shown  where  a  rope  would  stretch  to  such  an  extent 
that  it  had  to  be  taken  up  every  few  days  when  new,  and 
later  every  two  or  three  weeks— a  rope  under  these  condi- 
tions lasting  less  than  four  months.  Investigation  showed 
that  a  tension-weight  of  600  pounds  had  been  placed 
upon  the  rope,  which  was  of  f  inch  diameter.  A  weight 
of  50  pounds  was  substituted,  and  the  rope  has  since  been 
running  satisfactorily  for  three  years,  and  has  only  been 
taken  up  once  in  that  time. 

Cases  where  ropes  have  suddenly  broken  are  few  in  num- 
ber, the  risk  in  this  respect  being  reduced  to  a  minimum 
by  the  fact  that  any  defect  in  a  rope,  arising  either  from 
wear  or  other  cause,  will  show  itself  long  before  the  point 
of  danger  is  reached. 

The  ropes  by  which  the  power  is  transmitted  consist  of 
an  elastic  substance,  and  their  lightness,  elasticity,  and 
comparative  slackness  between  the  pulleys  are  highly  con- 
ducive to  their  taking  up  any  irregularity  that  may  occur 
in  the  motive  power. 

Their  quiet  working  and  convenience  in  application, 
much  more  so  than  wide  belts,  are  also  features  which 
caused  ropes  to  be  looked  upon  with  great  favor.  An- 
other reason  for  its  rapid  progress  in  England,  which  was 
considered  a  great  advantage  by  the  millowners,  was  the 
adoption  of  the  multiple  rope,  now  known  as  the  English 
system,  in  which  a  number  of  single  ropes  are -spliced  and 
run  side  by  side. 

The  entire  freedom  from  any  risk  of  a  breakdown  or 
stoppage  of  the  works  which  might  occur  with  gearing  was 
an  important  factor  in  replacing  the  latter  by  the  newer 
system;  the  working  stress  in  the  ropes  being  but  a  fraction 
of  their  breaking  strength,  any  signs  of  weakness  in  an  in- 
dividual rope  would  allow  it  to  be  removed,  and  the  engine 
run  with  the  remaining  ropes  until  a  convenient  opportu- 
nity offered  for  the  replacement  of  the  weak  member. 


One  of  the  great  advantages  of  rope-driving  over  gearing 
lies  in  the  steady  motion  produced,  but  this  has  been  at- 
tributed more  to  an  accidental  combination  of  heavy  fly- 
wheel and  high  velocity  than  to  any  inherent  advantage  in 
the  system  itself. 

In  spur  gears  diameters  of  20  to  24  feet  were  usual  in 
large  engines,  while  for  ropes  the  pulleys  are  25  to  35  feet 
in  diameter;  and,  whereas  the  gears  weighed  from  20  to  25 
tons,  we  find  rope  pulleys  used  for  the  same  work  weighing 
from  60  to  75  tons.* 

When  we  consider  the  speed  at  which  these  heavy  wheels 
are  run — from  3000  to  GOOO  feet  per  minute — it  is  not 
surprising  that  uniform  rotation  is  obtained;  and  whether 
it  be  that  the  energy  stored  in  the  moving  mass  prevents 
fluctuation,  or  whether  the  elasticity  and  other  properties 
of  the  rope  perform  the  same  office,  or — which  is  more 
reasonably  the  case — all  of  these  factors  act  together,  the 
truth  remains  that  steadier  running  and  a  greater  output 
are  now  obtained  with  rope-driving  than  was  formerly  the 
case  with  toothed  gearing. 

The  general  experience  is  not  altogether  in  favor  of  ropes, 
for,  while  the  advantages  of  smooth  running  and  easy 
handling  are  conceded,  it  is  also  acknowledged  that  the 
extra  weight  and  greater  width  of  pulleys  increase  the 
journal  friction  over  that  found  with  toothed  gearing,  and 
that  otherwise  a  greater  loss  of  power  occurs,  the  causes  of 
which  will  be  discussed  subsequently. 

*  The  Walker  Manufacturing  Co.  of  Cleveland  recently  made  four 
large  rope  pulleys  for  the  Broadway  Cable  Railway  Company,  New- 
York,  which  weighed  104  tons  each.  These  were  32  feet  in  diameter 
and  were  grooved  for  thirty-four  2-inch  ropes. 

Another  example  of  heavy  rope  wheel  is  given  in  London  Engin- 
eer, January  11,  1884.  This  wheel  was  made  by  Hick,  Hargraves  & 
Co  ,  Bolton,  England,  for  a  cotton  mill  in  India,  and  is  30  feet  iu 
diameter,  15  feet  face,  grooved  for  60  ropes  to  transmit  4000  horse- 
power. Its  weight  is  140  tons. 


6  ROPE-DRIVING. 

It  is  difficult,  however,  to  determine  the  relation  between 
the  power  absorbed  by  ropes  and  gears,  for  in  nearly  all 
cases  where  rope-driving  has  been  substituted  for  gears, 
other  changes  have  been  made  at  the  same  time,  or  the 
engines  were,  after  the  alteration,  driven  at  an  increased 
speed,  so  that  there  has  been  little  opportunity  for  com- 
parison. There  is,  however,  a  generally -accepted  opinion 
among  engineers  that  the  loss  in  rope-transmission  is  from 
5  to  10  per  cent  greater  than  with  gearing. 

Mr.  A.  G.  Brown*  states  that  in  the  older  cotton-mills  of 
England,  where  the  main  drives  are  by  gearing,  and  belting 
is  used  for  the  intermediate  and  machine  driving,  the  fric- 
tion of  the  engine,  shafting,  gearing,  and  belting  averages 
about  20  per  cent  of  the  whole  power,  the  engine  develop- 
ing at  full  loads  from  500  to  800  I.  H.  P. 

These  engines  were  compound  condensing,  and  consisted 
in  each  case  of  an  overhead  beam  engine  which  had  been 
converted  into  a  compound  by  the  addition  of  a  high-press- 
ure cylinder  between  the  crank  and  beam  centre. 

In  the  newer  mills,  for  doing  practically  the  same  work, 
and  where  the  main  and  some  of  the  intermediate  drives 
are  by  ropes,  the  friction  of  the  engine,  ropes,  shafting,  and 
belting  averages  23  to  25  per  cent,  of  the  whole  power,  the 
engines  developing  from  800  to  1500  I.  H.  P.  It  is  fair  to 
assume  that  the  newer  engines  have  at  least  no  greater  per- 
centage of  friction  than  the  older  ones,  except  that  due  to 
an  increased  journal  friction  attributable  to  the  larger 
journals  and  greater  weight  of  the  rope-pulleys  ;  also  the 
friction  of  the  intermediate  shafting  and  small  belts  should 
be  the  same  in  each  case  ;  therefore  it  is  reasonable  to  con- 
clude that  the  increase  of  power  absorbed  by  the  use  of 
rope-driving  is  chargeable  to  the  system  itself.  The  writer 
has  assumed  in  similar  cases  that  the  loss  at  the  engine  in 

*  American  MacJiinist,  July  21,  1888. 


ROPE-DRIVING.  7 

rope  driving  is  about  10  per  cent  of  the  indicated  horse- 
power, that  an  additional  10  per  cent  is  absorbed  by  the 
mill-shafting,  and  that  from  5  to  8  per  cent  may  be  at- 
tributed to  losses  in  the  rope  itself  due  to  resistance  to 
bending,  wedging  in  the  grooves,  differential  driving  effect, 
and  creep,  all  of  which  affect  the  loss  to  a  greater  or  less 
extent.  As  compared  with  the  above,  the  friction  of  shaft- 
ing and  engines  in  American  cotton-mills,  where  belting  is 
used  exclusively,  indicates  that  the  percentage  of  loss 
where  large  belts  are  employed  is  probably  a  trifle  less  than 
that  obtained  with  ropes  in  English  mills  ;  but  the  condi- 
tions of  practice  are  so  varied  that  it  is  difficult  to  compare 
the  two  systems  from  published  results  of  tests.  As  shown 
on  page  155  for  similar  installations,  the  loss  absorbed  in 
shaft-friction  will  not  materially  differ  in  the  two  systems ; 
for  large  transmissions  the  engine -friction  should  be  less 
with  ropes,  but  the  losses  in  the  ropes  themselves  due  to 
islip,  differential  driving,  bending,  creep,  and  other  causes, 
may,  without  special  precautions,  exceed  to  a  small  degree 
the  losses  due  to  the  belt. 

Mr.  J.  T.  Henthorn,  in  a  paper  read  before  the  American 
Society  of  Mechanical  Engineers,  states  that  the  friction  of 
the  shafting  and  engine  in  a  print- mill  should  not  exceed  19 
per  cent  of  the  full  power;  but  out  of  fifty-five  examples  of 
a  miscellaneous  character  which  he  has  tabulated  only  seven 
cases  are  below  20  per  cent,  20  vary  from  20  to  25  per 
cent,  fifteen  from  25  to  30  per  cent,  eleven  from  30  to  35  per 
cent,  and  two  above  35  per  cent.  We  note  that  the  greater 
number  lies  between  20  and  25  per  cent;  allowing  a  varia- 
tion of  5  per  cent  each  side  of  these  limits  we  shall  obtain 
values  from  which  a  fair  average  may  be  determined.  This 
will  include  those  cases  under  20  per  cent,  but  not  those 
over  30,  from  which  we  find  the  mean  loss  to  be  23.9  per 
cent  of  the  total  power. 

Mr.  Barrus,  speaking  of  this  subject,  quotes  eight  cases, 


8  ROPE- DRIVING. 

the  data  of  which  were  obtained  from  tests  made  by  him- 
self in  various  New  England  cotton-mills,  in  which  the 
minimum  percentage  was  18  and  the  maximum  25.7,  the 
total  average  being  22  per  cent. 

Mr.  Samuel  Webber  states  that  16  per  cent  of  the  total 
power  of  a  mill  is  sufficient  to  overcome  the  friction  of 
shafting  and  engine — 10  per  cent  for  the  shafting  and  6 
per  cent  for  the  engine.  But  in  this  estimate  Mr.  Webber 
does  not  include  the  loss  due  to  the  belts  running  upon 
loose  pulleys,  which  he  does  not  consider  to  be  part  of  the 
shafting,  as  they  are  not  so  running  while  the  machinery 
is  in  operation  ;  and  when  it  is  not,  they  may  be  thrown  off 
as  well  as  not,  except  for  convenience. 

He  further  estimates,  both  from  his  own  experience  and 
the  observations  of  others,  that  the  power  consumed  by  the 
machine  belts  on  the  loose  pulleys  in  a  large  cotton-mill 
is  about  8  per  cent  of  the  whole.  This  8  per  cent 
added  to  the  16  per  cent  loss  due  to  shafting  and  engine 
will  give  24  per  cent  of  the  total  power — a  result  which 
agrees  closely  with  the  average  values  given  above.* 

Considering  the  greater  loss  which  occurs  in  the  use  of 
ropes  and  belts  for  main  drives,  the  recent  revival  of  gear- 
ing for  this  purpose  in  England  has  much  in  its  favor. 
Of  this  Mr.  Geo.  Eichards  states :f  "The  advantages  of 
rope  transmission  for  main  drives  in  large  plants  would 
not  be  as  apparent  if  compared  with  modern  gearing. 
The  kind  of  gear  used  for  this  purpose  to-day  is  not 
the  rough  cast  gear  used  formerly,  whose  uneven  motion 
produced  a  rumbling  which  could  be  heard  a  mile  or  two 
from  the  mill.  The  present  gears  are  often  machine-cut, 
made  to  bear  equally  on  each  tooth,  and  with  a  contact 

*  See  "Dynamometers  and  the  Measurement  of    Power,"  John 
Wiley  &  Sons,  New  York. 

f  Richards's  "Mechanical  Progress,"  1891. 


ROPE-DRIVING.  9 

across  the  whole  face  of  the  gear,  causing  little  more  noise 
than  the  ropes. 

With  greater  speed  and  stronger  and  heavier  gears  steady 
running  is  insured,  and  by  using  properly-proportioned 
machine-cut  teeth  comparatively  little  noise  results.  The 
saving  of  space  is  also  an  important  factor  advanced  in 
favor  of  gearing."  At  the  present  time  steel  gears  30  feet  in 
diameter  are  being  made,  with  all  the  teeth  cut,  for  trans- 
mitting the  powers  of  mill-engines  in  the  Old  ham  district, 
while  in  this  country  machine-cut  gears  from  30  to  50  feet 
in  diameter  are  in  use. 

However,  although  such  gearing  may  be  very  superior 
to  the  former  slow-running  cast  gears,  it  is  questionable 
whether  it  can  ever  produce  the  same  steadiness  of  running 
which  is  so  largely  a  distinguishing  feature  of  rope-trans- 
mission. Any  shock  or  sudden  fluctuation  of  load  must 
necessarily  be  transmitted  through  the  gear  teeth,  whereas 
with  rope  transmission  such  shock  is  partially  absorbed  by 
the  more  or  less  elastic  ropes  and  subsequently  given  out 
by  their  recoil. 

For  this  reason  when  uniformity  of  speed  is  desired 
ropes  are  generally  to  be  preferred  to  gears,  even  when  the 
latter  are  working  under  their  most  advantageous  con- 
ditions. 


10  ROPE-DRIVING. 


CHAPTER  II. 

WHERE  ropes  have  been  used  to  replace  gearing  in  the 
English  mills  the  plan  adopted  has  been  to  put  in  a  new- 
grooved  fly-wheel,  or  to  place  grooved  segments  upon  the 
existing  fly-wheel,  when  the  speed  could  be  increased  suffi- 
ciently to  allow  of  a  limited  number  of  ropes  being  em- 
ployed, and  the  width  of  the  wheel-pit  was  also  sufficient 
for  the  purpose  ;  but  if  this  plan  could  not  be  adopted 
grooved  pulleys  were  put  on  the  intermediate  shaft,  and  the 
ropes  carried  to  the  different  stories  of  the  mill.  It  has 
sometimes  been  necessary  to  put  in  a  countershaft,  so  as  to 
gain  speed  and  obtain  a  sufficient  distance  between  the 
centres  of  the  shafts  on  which  the  pulleys  are  placed. 

Where  rope-driving  has  been  installed  in  new  factories 
special  provision  has  been  made  for  the  ropes,  and  we  find 
in  such  cases  rope-wells  or  chambers  built  in,  suitably  fitted 
with  platforms  and  staircases  to  give  access  to  pulleys  and 
bearings  on  the  various  shafts,  as  shown  in  Figs.  1  and  2. 

The  majority  of  drives  are  arranged  so  that  the  ropes  are 
horizontal  or  inclined  rather  than  vertical,  and  with  the 
driving  or  tight  side  of  the  rope  on  the  lower  side  of  the 
pulleys  ;  then,  when  transmitting  power,  the  two  sides  ap- 
proach each  other,  and  the  arc  of  contact  is  increased. 

An  additional  advantage  is  that  obtained  by  the  weight 
of  the  rope  acting  on  both  pulleys,  thus  allowing  a  low 
initial  tension  to  be  maintained.  This  does  not  hold  for 
short  distances  between  centres,  as  under  such  conditions 
the  weight  of  the  rope  adds  little  to  the  total  tension;  on 
the  other  hand,  where  the  distance  between  the  pulleys  in 


12  ROPE-DRIVING. 

vertical  drives  is  small,  the  relative  weight  of  the  rope  be- 
ing small  as  compared  with  its  tension,  there  will  be  little 
tendency  for  the  rope  to  leave  the  bottom  sheave. 

With  these  conditions  the  efficiency  of  vertical  drives 
will  approach  that  of  a  horizontal  or  inclined  arrangement 
of  ropes. 

The  manner  of  distribution  of  the  power  to  the  several 
floors  of  a  mill  is  shown  in  Fig.  1,  which  represents  a 
plant  designed  by  Messrs.  Lockwood  &  Greene  of  Boston 
for  the  Lanett  Cotton  Mills,  West  Point,  Ga.,  in  which 
1100  h.p.  is  delivered  from  the  engine  by  means  of  twenty- 
six  If-inch  ropes.  The  fly-wheel  is  26  feet  in  diameter,  and 
makes  60  revolutions  per  minute,  corresponding  to  which 
the  velocity  of  the  rim  is  4900  feet  per  minute.  As  will  be 
seen  from  the  figure,  the  driving-sheaves  are  placed  in  a 
well  in  the  middle  of  the  factory,  and  the  line-shafts  ex- 
tend to  the  right  and  left,  as  shown. 

There  is  no  line-shaft  on  the  second  floor,  as  the  various 
machines  may  be  driven  from  below. 

The  distribution  is  as  follows  : 

ROPE-DRIVES  IN  LANETT  COTTON  MILLS. 


Number 
of 
Ropes. 

Dia.  of 
Rope. 
Inches 

Dia.  of 
Pulley. 
Inches. 

Revolutions 
of  Pulley 
per  minute. 

Horse- 
power. 

1st  floor  

8 

If 

81 

231 

336 

3d  floor              .            ... 

7 

1£ 

62 

302 

294 

4th  floor     

11 

If 

62 

302 

462 

A  similar  plant,  also  designed  by  Messrs.  Lockwood  & 
Greene,  has  been  recently  erected  at  the  Naumkeag  Cotton 
Mills,  Salem,  Mass.,  in  which  1800  horse-power  is  distrib- 
uted to  five  floors  by  means  of  forty-one  If -inch  manilla 
ropes.*  The  fly-wheel  is  26  feet  in  diameter,  by  about  9£ 

*  Power,  March  1895. 


ROPE-DRIVING. 


13 


OF  THE 

XTNIVERSITt" 


14 


ROPE-DRIVING. 


feet  face,  and  weighs  150,000  pounds.  The  velocity  of  the 
ropes  is  the  same  as  in  the  previous  instance,  namely,  4900 
feet  per  minute.  The  distribution  is  as  follows: 


Number  of 
Ropes. 

Dia  of 
Rope. 

Speed  of  Rope. 

Horse- 
power. 

1st  floor  

13 

1  ji  inches 

4900  ft  p  m 

58  T 

2d       "    

14 

fjqn 

3d       ".-..;. 

4th     " 

4 
5 

" 

« 

180 
99*; 

5th     "    .      .   . 

5 

t( 

tt 

99ni 

Total  

41 

1845 

A  system  of  main  driving-geap  designed  and  erected  by 
John  Musgrave  &  Sons  at  the  Atlas  Mills,  Bolton,  Eng.,  is 
shown  in  Figs.  2  and  3.  This  mill  is  300  feet  long  by  135 
feet  wide,  with  a  shed  on  one  side  325  feet  long  by  45  feet 
wide,  and  contains  84,000  spindles. 

The  engines  are  tandem  compound,  24  and  46  by  6  ft. 
stroke,  and  run  at  50  revolutions  per  minute.  The  average 
horse-power  is  1050.  The  rope-wheel  is  32  feet  in  diameter 
and  is  grooved  for  thirty-two  11 -inch  cotton  ropes,  which 
run  at  5026  feet  per  minute. 

The  arrangement  of  shaft  is  as  follows :  On  the  ground- 
floor  there  are  five  lines  of  shafting,  the  main  shaft  being 
driven  from  the  rope-drum  by  means  of  ten  ropes  If  inches 
diameter  running  on  a  pulley  9  feet  4|  inches  in  diameter, 
which  runs  170  revolutions  per  minute. 

On  the  main  shaft,  close  to  the  wall  of  the  rope-well,  is 
a  pulley  6  feet  diameter,  grooved  for  four  If-inch  ropes, 
which  drives,  through  a  similar-sized  pulley,  the  line-shaft 
on  the  right.  These  ropes  run  at  a  velocity  of  3205  feet 
per  minute;  also  on  the  main  shaft,  but  on  the  opposite 
side  of  the  rope-well,  is  another  pulley,  8  feet  diameter, 
grooved  for  six  If-inch  ropes,  driving  on  to  a  pulley  64^ 


ROPE-DRIVING. 


15 


16  KOPE-DKIVI^G. 

inches  diameter  on  the  first  line-shaft  to  the  left  of  the 
main  shaft.  This  shaft  runs  250  revolutions  per  minute, 
corresponding  to  which  the  speed  of  the  ropes  driving  it  is 
4274  feet  per  minute. 

The  second  line-shaft  on  the  left  is  driven  from  the  first 
one  by  means  of  a  pair  of  pulleys  4  feet  diameter,  grooved 
for  five  If -inch  ropes.  This  second  shaft  runs  250  revolu- 
tions per  minute,  and  the  ropes  driving  it  have  a  velocity 
of  3140  feet  per  minute. 

The  second  line-shaft,  mentioned  above,  drives  the  line- 
shaft  in  the  shed  by  means  of  a  pair  of  pulleys  4  feet  di- 
ameter, grooved  for  four  li-inch  ropes,  driving  the  counter- 
shaft shown  on  plan,  on  which  is  a  pulley  3  feet  4  inches 
diameter,  grooved  for  five  H-inch  ropes,  driving  on  a  3-ft. 
pulley  on  the  shed  line-shaft.  These  ropes  have  a  velocity 
of  2640  feet  per  minute  and  give  to  the  shaft  278  revolu- 
tions per  minute. 

All  of  the  shafts  described  are  on  the  ground -floor  of  the 
mill.  Of  the  shafts  above  this,  on  the  next  two  floors,  the 
line-shafts  each  have  pulleys  6  feet  in  diameter,  grooved  for 
seven  If-inch  ropes,  driven  from  the  main  rope-drum. 
These  shafts  run  at  266  revolutions  per  minute.  The 
shaft  on  the  upper  floor  also  runs  266  revolutions  per  min- 
ute and  is  driven  from  the  main  rope-drum  through  a  pul- 
ley 6  feet  in  diameter  grooved  for  eight  If-inch  ropes. 

The  distance  from  the  centre  of  upper  shaft  to  the  cen- 
tre of  crank-shaft  is  89  feet,  and  the  length  of  each  rope  re- 
quired for  this  drive  is  about  250  feet. 

Another  arrangement  of  rope-drive  for  cotton-mills  is 
shown  in  Fig.  4,  which  represents  a  section  through  the 
engine-room  at  the  thread-mills  of  the  Nevsky  Cotton- 
Spinning  Co.,  St.  Petersburg.*  In  this  drive  there  is  no 
rope-chamber,  as  the  whole  of  the  rope  gear  is  contained 

*  John  Musgrave  &  Sons,  engineers. 


ROPE-DRIVING.  17 

in  the  engine-room  situated  in  the  centre  of  the  mill,  which 
is  680  feet  long  and  90  feet  wide. 

There  is  a  short  staircase  from  the  engine-room  floor  to 
the  first  landing,  and  the  landings  above  this  are  reached 
by  an  ornamental  spiral  stairway  as  shown. 

There  are  two  shafts,  one  from  each  side  of  the  engine- 
room  ;  these  are  driven  by  a  pair  of  right-  and  left-hand 
tandem  compound  engines,  30  and  52  by  6  ft.  stroke,  run- 
ning at  50  revolutions  per  minute. 

The  average  power  developed  by  each  engine  is  1100 
horse-power. 

The  rope-drums  are  each  30  feet  in  diameter  and  weigh 
62  tons;  these  are  grooved  for  twenty-eight  If  cotton  ropes, 
which  run  at  4700  feet  per  minute. 

The  first-and  second-floor  shafts  make  300  revolutions 
per  minute,  and  are  each  driven  by  nine  If -inch  ropes  from 
the  rope-drum  running  over  pulleys  5  feet  in  diameter  on 
the  shafts. 

The  shafts  on  the  third  and  fourth  floors  run  200  revo- 
lutions per  minute,  and  are  driven  by  five  ropes,  each  If 
inches  diameter,  running  over  7  feet  6  inch  pulleys  on  the 
shafts. 

In  each  of  the  two  upper  rooms  there  is  a  second  line- 
shaft,  driven  from  the  main  line-shaft  on  each  floor  by 
means  of  54-inch  pulleys,  grooved  for  four  l^-inch  ropes, 
which  have  a  velocity  of  2826  feet  per  minute.  The 
distance  from  the  centre  of  the  upper  shaft  to  the 
centre  of  the  crank-shaft  of  engine  is  56  feet  6  inches. 
This  is  a  short  drive  for  a  mill  of  this  size;  in  fact,  all  of 
the  drives  are  short,  the  lower  one  especially  so,  being  only 
80  feet  between  centres,  the  peculiar  arrangement  of  the 
engine-room  not  admitting  of  a  greater  length;  but  the 
plant  is  said  to  work  extremely  well. 

In  these  examples  the  multiple-rope  system  is  used,  each 
wind  consisting  of  a  separate  rope  stretched  around  the  fly- 


18 


ROPE-DRIVING. 


wheel  and  its  individual  shaft-pulley,  then  spliced.  The 
degree  of  tightness  will  depend  upon  the  material  of  the 
rope  and  the  amount  of  tension  in  the  slack  part  necessary 
for  adhesion.  In  the  majority  of  cases  the  initial  tension 
is  very  small  compared  with  the  strength  of  the  rope — es- 
pecially so  where  the  horizontal  distance  between  driving 
and  driven  pulleys  is  great,  as,  under  such  conditions,  the 
tension  in  the  slack  side  due  to  the  weight  of  rope  in  the 
hanging  catenary  is  often  sufficient  to  prevent  slipping;  a 

slack  upper  rope  in  horizontal 
or  inclined  drives  will  also 
increase  the  arc  of  contact, 
thereby  increasing  the  grip 
of  the  rope. 

This  is  shown  in  Fig.  5, 
which  represents  two  pulleys 
of  equal  diameter,  arranged, 
as  in  the  upper  figure,  to 
drive  with  the  slack  side  uppermost,  and,  in  the  lower 
figure,  with  the  slack  side  below. 

The  difference  in  the  arc  of  contact,  as  shown  in  the 
figures,  is  60  degrees,  which  would  under  similar  con- 
ditions, with  a  velocity  of  4000  feet  per  minute,  produce  a 
difference  of  over  25  per  cent  in  the  amount  of  horse-power 
transmitted  by  the  two  ropes  under  the  usual  working 
tension. 

New  cotton  ropes  are  often  stretched  as  taut  as  possible 
on  account  of  their  extensibility,  as  they  will  soon  become 
slack  enough  for  good  working,  and  may  even  have  to  be 
respliced  before  becoming  permanently  set. 

It  is  the  practice  of  some  engineers  to  strain  both  manilla 
and  cotton  ropes  as  much  as  possible  and  unite  the  ends 
with  a  temporary  short  splice  when  first  put  over  the  pul- 
leys; after  running  a  few  days  a  permanent  stretch  is  given 
to  the  rope,  which  is  then  respliced  with  a  long  splice,  the 


FIG.  5. 


ROPE-DRIVING.  19 

strain  on  the  rope  being  very  much  reduced  in  this  latter 
case. 

The  splice  in  a  transmission  rope  is  not  only  the  weakest 
part  of  the  rope,  but  is  the  first  to  fail  when  the  rope  is 
worn  out.  If  the  joint  is  not  strong  the  rope  will  fail  by 
breakage  or  pulling  out  of  the  splice,  the  projecting  parts 
will  wear  on  the  pulleys,  and  the  rope  fail  from  the  cutting 
off  of  the  threads.  Formerly  much  trouble  was  experienced 
in  this  way  on  account  of  improper  splicing.  One  form  of 
joint,  according  to  Cromwell,*  was  made  by  pressing  the 
ropes  firmly  together  and  winding  about  with  stout  small 
rope.  The  spliced  part  is  taken  as  long  as  possible  in  order 
to  bend  properly  over  the  pulleys  and  give  the  required 
strength.  As  this  form  of  joint  made  the  rope  larger  in 
diameter  at  the  splice,  the  effect  produced  was  to  run 
faster  when  passing  over  the  driving-sheave  and  slower 
over  the  follower;  the  resulting  motion  was  very  irregular, 
and  the  wear  at  the  splice  rapidly  destroyed  the  rope. 

A  very  simple  splice  is  sometimes  used  with  rope-driving 
formed  by  opening  out  the  ends  of  the  rope  for  12  or  15 
inches  and  tying  together  the  individual  rope-yarns  one 
by  one,  allowing  the  ends  to  lie  straight,  and  serving  the 
whole  with  spun  yarn. 

Similar  joints  wrapped  with  raw-hide  belt-lacing  give  a 
very  smooth  splice  which  lasts  well. 

Some  engineers  favor  a  short  splice,  in  that  it  is  easily 
made  and  holds  well,  and  offers  a  lesser  length  of  enlarged 
portion  for  surface  contact  with  the  pulley. 

If  properly  made,  however,  there  need  be  no  enlarged 
portion,  and  since  a  long  splice  is  stronger  we  find  such 
joints  preferred  in  most  cases. 

There  are  several  kinds  of  long  splices  varying  in  length 
from  GO  to  80  diameters  of  rope,  but  the  one  which  seems 

*  J.  H.  Cromwell,  "  Belts  and  Pulleys,"  John  Wiley  &  Sons. 


20  ROPE-DRIVING. 

to  give  the  best  results  in  practice  is  the  "  English  splice/' 
directions  for  which  are  given  in  various  trade  publications. 
The  successive  operations  for  splicing  a  If -inch  rope  by 
this  method  are  as  follows :  * 

1.  Tie  a  piece  of  twine,  9  and  10,  Fig.  6,  around  the  rope 
to  be  spliced  about  six  feet  from  each  end.     Then  unlay 
the  strands  of  each  end  back  to  the  twine. 

2.  Butt  the  ropes  together  and  twist  each  corresponding 
pair  of  strands  loosely,  to  keep  them  from  being  tangled, 
as  shown  at  (a),  Fig.  6. 

3.  The  twine  10  is  now  cut,  and  the  strand  8  unlaid  and 
strand  7  carefully  laid  in  its  place  for  a  distance  of  four 
and  a  half  feet  from  the  junction. 

4.  The  strand  6  is  next  unlaid  about  one  and  a  half  feet 
and  strand  5  laid  in  its  place. 

5.  The  ends  of  the  cores  are  now  cut  off  so  they  just 
meet. 

6.  Unlay  strand  1  four  and  a  half  feet,  laying  strand  2 
in  its  place. 

7.  Unlay  strand  3  one  and  a  half  feet,  laying  in  strand  4. 

8.  Cut  all  the  strands  off  to  a  length  of  about  twenty 
inches,  for  convenience  in  manipulation.     The  rope  now 
assumes  the  form  shown  in  (b),  with  the  meeting-points  of 
the  strands  three  feet  apart. 

Each  pair  of  strands  is  now  successively  subjected  to  the 
following  operations : 

9.  From  the  point  of  meeting  of  the  strands  8  and  7 
unlay  each  one  three  turns;  split  both  the  strand  8  and 
the  strand  7  in  halves,  as  far  back  as  they  are  now  unlaid, 
and  the  end  of  each  half  strand  "  whipped "  with  a  small 
piece  of  twine. 

10.  The  half  of  the  strand  7  is  now  laid  in  three  turns, 
and  the  half  of  8  also  laid  in  three  turns.     The  half  strands 

*  From  "  Manilla  Rope,"  C.  W.  Huut  Co.,  New  York. 


HOPE  DKIVIKG. 


21 


FIG.  6. — SPLICE  FOR  If -INCH  4- STRAND  ROPE. 


22  ROPE-DRIVING. 

now  meet  and  are  tied  in  a  simple  knot  11,  (c),  making  the 
rope  at  this  point  its  original  size. 

11.  The  rope  is  now  opened  with  a  marlinspike,  and  the 
half  strand  of  7  worked  around  the  half  strand  of  8  by  pass- 
ing the  end  of  the  half  strand  through  the  rope,  as  shown, 
drawn  taut,  and  again    worked  around  this  half  strand 
until  it  reaches  the  half  strand  13  that  was  not  laid  in. 
This  half  strand  13  is  now  split,  and   the  half  strand   7 
drawn  through  the  opening  thus  made,  and  then  tucked 
under  the  two  adjacent  strands,  as  shown  in  (d). 

12.  The  other  half  of  the  strand  8  is  now  wound  around 
the  other  half  strand  7  in  the  same  way.     After  each  pair 
of  strands  has  been  treated  in  this  manner,  the  ends  are 
cut  off  at  12,  leaving  them  about  four  inches  long.     After 
a  few  days'  wear  they  will  draw  into  the  body  of  the  rope 
or  wear  off,  so  that  the  locality  of  the  splice  can  scarcely 
be  detected. 

For  a  three-strand  rope  of  the  same  size  the  foregoing 
method  is  slightly  modified.  After  tying  the  twine  9  and 
10  around  the  rope  about  6  feet  from  each  end,  unlay  the 
strands  back  to  the  twine,  bring  the  butts  together,  and, 
as  in  Fig.  7,  twist  the  corresponding  strands  loosely  together. 
Now  cut  twine  10,  and  unlay  strand  8  for  a  distance  of 
four  and  a  half  feet  from  the  junction,  and  lay  in  strand 
7.  Unlay  strand  1  four  and  a  half  feet,  lay  in  strand  2, 
and  cut  all  the  strands  off  to  a  length  of  about  20  inches, 
as  before  explained  for  convenience  in  handling.  The 
splice  now  assumes  an  appearance  similar  to  (&)  with  the 
exception  that  there  are  only  three  meeting-points  of  the 
strands,  and  these  are  4J  feet  apart. 

Each  pair  of  strands  is  now  subjected  to  the  series 
of  operations  described  for  the  4-strand  splice  in  steps  9 
to  12  inclusive. 

In  splicing  a  Lambeth  cotton  rope  the  operation  is  modi- 
fied to  a  still  greater  extent, 


ROPE-DRIVIXG.  22a 

Although  considered  as  a  troublesome  rope  to  splice,  the 
following  instructions,*  if  carefully  followed,  will  enable 
one  to  make  an  excellent  joint  without  difficulty. 

1.  Tie  a  piece  of  twine  9  and  10  around  the  rope  to  be 
spliced  about  6  feet  from  each  end.  .  Then  unlay  two 
strands  together  of  each  end  back  to  the  twine.  Butt  the 


FIG.  A. 

ropes  together  and  tie  each  set  of  strands  temporarily,  as 
shown  at  Fig.  A. 

2.  The  twine  10  is  now  cut,  and  the  strands  6  and  8 
unlaid  together,  and  the  strands  5  and  7  carefully  laid  in 
their  places  together  for  a  distance  of  18  inches.  Then 
unlay  strands  6  and  8,  also  5  and  7,  and  tie  strands  6  and 
5  together  temporarily.  Next  unlay  strand  7  and  lay  in 
strand  8  in  its  place  for  a  distance  of  3  feet  from  strands  6 

*  Prepared  for  this  work  by  the  Manufacturers'  Engineering  Co., 
Boston,  Mass, 


Z'-0  ROPE-DRIVING. 

and  5.  Theii  tie  strands  8  and  ?  temporarily.  Next  cut 
off  the  ends  of  the  core  so  that  they  will  butt  together. 
Strands  1,  2,  3,  and  4  are  next  laid  in  the  same  manner  as 


FIG.  B 

strands  5,  6,  7,  and  8,  but  in  the  opposite  direction.  Care 
must  be  taken  to  keep  the  turns  in-  the  strands,  or  otherwise 
they  will  be  soft  and  bulky.  Next  cut  off  all  the  strands 


FIG.  C. 


to  a  length  of  about  24  inches,  for  convenience  in  handling. 
At  this  point  the  splice  should  be  as  shown  in  Fig.  B, 


llOPE-DRtVIN-G.  22tf 

The  tension  strand  of  a  Lambeth  cotton  rope  is  the  soft 
white  yarn  running  through  the  centre  of  the  strand, 
and  is  called  the  tension  strand  through  its  having  to 
bear  the  strain  put  upon  the  rope  in  the  transmission  of 
power. 

The  friction  bands  of  a  Lambeth  rope  are  the  twisted 
outside  yarns  which  are  tubed  around  the  tension  strands 
to  protect  them  from  wear  and  contact  in  the  grooves  of 
the  pulley. 

3.  Take  strand  2  and  unlay  it  two  turns  and  remove  the 
ten  friction  bands,  then  lay  in  tension  strand  2  back  again 


FIG.  D. 


one  turn,  split  out  J  of  tension  strand  2  and  lay  in  the  re- 
maining |  of  tension  strand  2  for  one  turn.  This  will 
bring  it  to  its  former  position.  Remove  the  ten  friction 
bands  from  strand  1,  and  tie  tension  strand  1  and  }  of 
tension  strand  2  in  a  simple  knot.  At  this  point  of  the 
knot  the  rope  will  be  its  original  diameter,  as  shown  in 
Fig.  0. 

4.  Divide  the  friction  bands  removed  from  strand  1  in 
two  parts,  and  take  J  of  tension  strand  2,  put  it  between 
the  two  parts  and  over  tension  strand  1  and  through  the 


HOPE-DRtVIKG. 

centre  of  the  rope  with  the  marlinspike.  Next  take  ten- 
sion strand  1  and  work  it  around  the  }  of  tension-strand  2 
in  the  manner  as  shown  at  Fig.  D. 

5.  Draw  it  taut  and  continue  to  work  it  around  J  of  ten- 
sion strand  2  until  it  reaches  the  £  of  tension  strand  2; 
at  this  point  £  of  tension  strand  1  must  be  removed,  and 
continue  to  work  f  of  tension  strand  1  around  tension 
strand  2  until  it  reaches  the  friction  bands  removed  from 
strand  2;  divide  these  friction  bands  in  two  parts,  and  take 
f  of  tension  strand  1,  put  it  between  the  two  parts  and 


FIG 


over  tension  strand  2,  and  through  the  centre  of  the  rope 
with  the  marlinspike.  Next  take  the  quarter  of  tension 
strands  1  and  2  and  pass  them  through  the  centre  of  the 
rope  on  opposite  sides  with  the  marlinspike.  Then  half 
of  the  friction  bands  should  be  passed  through  the  centre 
of  the  rope  at  each  end  with  the  spike.  At  this  point 
the  splice  is  complete,  with  the  exception  of  cutting  oft' 
the  ends,  and  should  be  as  shown  at  Fig.  E. 

G.  The  strands  3,  4,  5,  6,  7,  and  8  should  be  worked  in 
the  same  manner  as  1  and  2. 

Instead  of  using  the  ordinary  marlinspike  it  will  be 
found  very  convenient  to  drill  out  the  body  as  shown  in 


OP  THE 

UNIVERSITY 


Fig.  8,  leaving  only  a  thin  shell  four  or  five  diameters 
deep. 

By  inserting  the  end  of  a  strand  iii  the  bore  of  the  mar- 


.  7.— ROPE-SPLICING. 


linspike  the  latter,  with  the  strand,  may  be  passed  through 
and  around  the  other  strands  as  desired  with  much  less 
trouble  than  ordinarily  attends  the  operation. 


FIG.  8. — IMPROVED  FORM  OF  MARLINSPIKE. 

For  small  braided  ropes  which  cannot  be  spliced,  a 
very  convenient  method  of  joining  the  ends  is  by  the 
use  of  copper  ferrules  as  shown  in  Fig.  9,  which  repre- 


FIG.  9.— COUPLING  FOR  BRAIDED  ROPE. 

sents  a  form  of  joint  devised  by  Mr.  B.  Frank  Barnes, 
Rookford,  Til. 


24  ROPE-DRIVING. 

Samples  of  tins  splice  were  furnished  by  Mr.  Jo^. 
Burnett,  which  showed  an  efficiency  of  about  85  per  cent. 
Thus  in  two  samples  the  average  breaking  strength  of  the 
rope  was  380  and  375  pounds  respectively,  while  the  splice 
pulled  out  under  a  strain  of  320  pounds. 

In  this  case  the  rope  was  a  f-inch  braided  cotton  cord 
which  had  been  in  use  about  three  years.  The  coupling 
consists  of  a  piece  of  copper  tubing  1  inch  long,  into  one 
end  of  which  the  rope  is  inserted  about  half  way.  A 
groove  is  then  compressed  around  the  tube  and  rope,  by 
means  of  a  special  tool;  the  open  end  of  the  tube  is  then 
filled  with  sealing-wax  and  heated  until  the  wax  boils, 
then  the  other  end  of  the  rope  js  inserted,  and  the  tube 
compressed.  The  melted  wax  fills  the  end  of  the  rope, 
making  a  solid  joint  between  the  shoulders.  With  the 
large  pulleys  adopted  (22  to  48 ;  inches  in  diameter)  no 
trouble  is  experienced,  and  the  ropes  last  from  two  to 
three  years,  but  the  copper  ferrules  are  changed  about 
every  four  months. 

Wood  pulleys  are  used,  and  the  grooves  are  filled  with 
leather.  Some  of  these  ropes  run  as  high  as  5300  feet  per 
minute.  All  the  ropes  are  operated  on  the  American  or 
continuous  wind  system. 


ROPE-DRIVING.  25 


CHAPTER  III. 

A  GOOD  exam  pie  of  this  system  of  rope  transmission  is 
shown  in  Fig.  10,  which  represents  a  plant  designed  by 
Mr.  T.  Spencer  Miller,  and  erected  for  the  Western  Electric 
Company,  New  York,  by  the  Link  Belt  Machinery  Co. 
In  this  case  vertical  ropes  are  used,  which  are  arranged  to 
transmit  the  power  of  two  200-h.  p.  Russell  engines, 
cylinders  18  by  27  inches,  making  125  revolutions  per 
minute;  fly-wheels  10  feet  diameter,  each  turned  with 
eight  grooves  for  1^-inch  rope.  The  ropes  are  of  raw- 
hide and  wound  continuously  around  the  pulleys.  As  the 
rope  leaves  the  fly-wheel  at  the  left-hand  side  it  runs  over 
an  idler,  and  from  thence  to  a  tension-pulley,  or  tightener, 
which  is  suspended  in  such  a  manner  as  to  be  drawn  back 
by  the  weights,  as  shown.  The  arms  which  support  the 
tighteners  are  hung  from  rollers,  which  are  grooved  to  fit 
the  surface  of  a  section  of  extra  heavy  wrought-iron  pipe, 
upon  which  they  roll.  From  this  tightener  the  rope 
passes  direct  to  the  right-hand  groove  of  the  pulley  on  the 
main  shaft  above  the  engine,  the  tightener-pulley  being 
inclined  sufficiently  to  make  the  bottom  come  in  line  with 
the  left-hand,  while  the  top  comes  in  contact  with  the 
right-hand  groove. 

The  main  pulleys  which  drive  the  shaft  above  the 
engines  are  mounted  upon  and  keyed  to  sleeves  10  inches 
diameter,  which  extend  out  on  each  side  far  enough  to 
form  journals,  by  which  they  are  supported  in  pedestals 
independently  of  the  shaft  (see  Fig.  33).  Through  the 
sleeve  is  a  hole  considerably  larger  than  the  shaft  which 
passes  through  them  and  which  is  supported  by  separate 


26 


ROPE-DRIVING. 


FIG.  10. — AMERICAN  OR  WOUND  SYSTEM 
OF  HOPE  TRANSMISSION, 


ROPE-DRIVING.  27 

pedestals.  One  end  of  each  sleeve  is  so  formed  as  to 
make,  in  connection  with  a  sliding  collar  which  is  on  the 
shaft,  a  positive  interlocking  clutch,  which  can  be  thrown 
in  or  out  by  a  lever.  In  this  way  both  engines  may  be 
working  at  the  same  time,  or  the  shaft  may  be  run  by 
either  engine  alone,  the  other  pulley  standing  and  impos- 
ing no  friction  upon  the  shaft.  All  the  bearings  of  this 
shaft  are  adjustable  laterally  by  set-screw  and  vertically  by 
wedges.  The  other  large  pulleys  upon  this  shaft  are 
driven  by  friction-clutches,  and  are  used  for  driving  dyna- 
mos, each  pulley  driving  two  dynamos  arranged  tandem, 
one  belt  running  over  the  other.  This  shaft  runs  at  220 
revolutions  and  is  44  inches  diameter.  At  one  end  of  the 
main  shaft  is  a  pulley  having  twelve  grooves,  in  which  run 
two  ropes  side  by  side  to  the  top  of  the  building  and 
around  the  various  pulleys  down  again.  Either  of  these 
ropes  is  calculated  to  be  amply  strong  for  the  work,  but 
two  are  used  to  avoid  the  necessity  for  stopping  should 
one  break.  Each  of  them  winds  three  times  around  the 
pulleys,  thus  giving  six  driving-strands.  To  avoid  crowd- 
ing, the  tightener  for  one  of  these  is  placed  upon  the  floor 
above  the  other.  The  ropes  pass  from  the  main  pulley 
three  times  around  the  pulley  above  and  then  go  to  the  upper 
floors,  as  indicated.  From  the  shafts,  the  wood-working 
machinery,  blowers  for  the  foundry,  and  some  of  the  eleva- 
tors are  operated;  the  other  lines  being  used  for  driving 
light  machine  tools,  such  as  are  used  in  making  electrical 
apparatus. 

Each  floor  has  a  cut-off  coupling,  which  is  so  arranged 
that  in  case  of  accident  it  can  be  cut  off  at  a  minute's 
notice,  or  when  running  overtime  any  floor  can  be  cut  off. 
thus  saving  the  cost  of  running  any  more  machinery  than 
is  necessary. 

The  use  of  the  tension -carriage  plays  an  important  part 
in  the  American  system  of  rope  transmission.  As  usually 


28 


ROPE-DRIVING. 


made  it  is  automatic  in  its  operation  and  so  weighted  as 
to  give  a  constant  tension  to  the  rope,  as  indicated  in 
Fig.  11.  In  this  arrangement  an  initial  tension  is  given 


FIG.  11. — AUTOMATIC  TIGHTENER  FOR  ROPE  TRANSMISSION. 

and  maintained  by  the  automatic  tension-carriage,  which 
is  free  to  move  backward  or  forward  on  a  horizontal  track 


FIG.  12.— TENSION-CARRIAGE. 


as  the  load  changes  or  the  rope  stretches,  always  taking 
up  the  slack  and  maintaining  the  proper  tension.  An- 
other form  of  tension -carriage  is  that  shown  in  Fig.  12. 


ROPE-DRIVING. 


29 


In  tliis  case  the  carriage  is  mounted  on  gas-pipe  or  solid 
shafting,  and  is  provided  with  ball  bearings  arranged  with 
cast  pockets  so  that  the  balls  are  allowed  to  circulate 
lengthwise  of  the  bearings. 

An  arrangement  of  light  angle-iron  tracks  supporting  a 
four-wheeled  carriage  is  used  to  a  considerable  extent  and 
makes  an  excellent  tightener  where  it  can  be 
employed,  as  it  is  cheap  and  readily  set  up. 

It  is  obvious  that  vertical  tensions  may  be 
arranged  in  a  similar  manner  to  those  shown 
in  Figs.  11  and  12.  In  such  cases  the  weight 
may  be  suspended  directly  from  the  carriage  or 
even  the  sheave,  as  in  Fig.  13,  or  it  may  be  led 
off  in  any  desired  direction  either  above  or 
below  the  tightener-pulley.  The  tightener 
pulley  is  often  inclined  from  the  vertical,  so 
that  its  projection  is  equal  to  the  width  of  the 
driving  pulley,  in  which  case  it  not  only  serves 
to  maintain  a  constant  tension  in  the  rope,  but  FIG.  13. 
it  thus  acts  as  a  guide  to  conduct  the  rope  from  one  groove 
to  another. 


FIG.  14. — ROPE-TIGHTENER. 


A  modification  of  the  usual  belt-tightener  is  sometimes 
used  for  rope  drives,  as  shown  in  Fig.  14,  but  this,  it  will 


30  ROPE-DRIVING. 

be  noticed,  is  positive  in  action  and  is  only  used  to  increase 
the  tension  on  the  ropes  as  the  latter  become  extended 
with  use. 

In  the  same  way  the  tightener  shown  in  Fig.  15  *  is 
used  to  take  up  any  slack  that  may  occur  in  the  rope. 

In  this  case  the  tightener-pulley  T  is  mounted  on 
a  standard  free  to  slide  in  the  bottom  guides  G.  A 
weighted  lever  L  is  connected  to  the  pinion-shaft  S  by 
means  of  a  ratchet-wheel  and  pawl  (not  shown  in  the 
figure).  On  the  other  end  of  this  shaft  and  rigidly  con- 
nected to  it  a  pinion  meshes  with  the  rack  11  upon  one  of 


FIG.  15. — DYBLIE'S  ROPE-TIGHTENER  FOR  DYNAMOS. 

the  guides  and  causes  the  standard  and  tightener-pulley 
to  move  along  the  base  and  thus  automatically  take  up  the 
slack  as  it  occurs.  A  detent,  D,  in  the  carriage  catches 
in  the  teeth  of  the  rack  and  prevents  the  tightener  from 
slipping  back.  It  is  evident  that  any  form  of  adjuster 
which  will  not  allow  the  tension-pulley  to  move  in  both 
directions — either  back  and  forth  or  up  and  down— does 
not  maintain  a  uniformity  of  tension;  as  the  humidity  in 
the  air  may  cause  a  rope  to  shrink  very  materially  in  a 
short  time  the  tension  on  the  rope  will  be  greatly  increased 
if  the  tightener-pulley  is  prevented  from  moving  in.f 

*  Patented  by  J.  A.  Dyblie,  Jan.  14,  1890. 

f  Mr.  Louis  I.  Seymour  states  that  in  a  certain  out-door  drive  at 


ROPE-DRIVING. 


31 


It  is  a  well-recognized  fact  that  atmospheric  changes 
affect  the  length  of  a  rope,  which  in  the  presence  of  mois- 
ture always  contracts.  Experiments  have  shown  that  a 
dry  hemp  rope  25  feet  long  will  shrink  to  24  feet  upon  be- 


FIG.  16. — TENSION-CARRIAGE  WITH  VARIABLE  ARM. 

ing  wet.*  It  is  for  this  reason  that,  in  addition  to  taking 
np  the  slack  caused  by  variation  in  load,  provision  should 

the  Plymouth  Cordage  Company's  Works  in  which  l£-inch  rope  is 
used  to  transmit  125  h.  p.  the  tension  carriage  is  drawn  in  about 
eight  feet  by  the  shrinkage  of  the  ropes  during  a  severe  storm  ;  that 
is,  the  rope  is  shortened  about  sixteen  feet.  The  total  length  of  rope 
in  this  case  is  approximately  1600  feet,  so  that  the  contraction  is  thus 
about  one  per  cent  of  its  total  length.  The  rope  used  was  of 
superior  quality  manilla,  laid  with  plumbago  and  tallow,  otherwise 
the  shrinkage  would  have  been  greatly  in  excess  of  the  amount 
stated. 

*  Indian  En.r/ineer,  1888. 


32  EOPE-D  RIVING. 

be  made  for  maintaining  a  proper  tension  in  the  rope 
when  the  latter  is  variable  in  its  length,  due  either  to 
atmospheric  changes,  or  permanent  elongation  as  the  rope 
loses  its  elasticity.  This  variation  in  length  is  particularly 
noticeable  in  rope-drives  which  are  subjected  to  exposure 
from  the  weather. 

An  arrangement  sometimes  adopted  with  horizontal  car- 
riages as  a  substitute  for  pulley  and  hanging  weight  is 
shown  in  Fig.  16.  In  this  arrangement  the  tail-rope, 
usually  of  wire,  is  wrapped  around  and  secured  to  a 
grooved  pulley-sheave  keyed  to  a  shaft  which  is  fixed  in 
position  but  free  to  rotate.  A  weighted  lever  is  secured 
to  the  shaft  by  means  of  a  set-screw  and  maintains  a  ten- 
sion on  the  rope  by  virtue  of  its  moment.  It  will  readily 
be  seen  that  this  tension  will  not  be  constant,  for  the 
effective  lever-arm  of  the  weight,  and  hence  the  pull  on 
the  tail-rope,  will  vary  with  the  position  of  the  lever;  thus 
in  its  normal  position  with  the  lever  horizontal  the  tension 

PR 

in  the  tail-rope  will  be  T  —  -  — ;  but  if  the  tension-car- 
riage moves  either  in  or  out  on  its  guides  the  lever  will 
assume  a  new  position  as  shown  in  dotted  lines,  in  which 

75  ~nf 

case  the  tension  will  now  be  T  =  -  — . 

As  the  initial  tension  which  gives  adhesion  to  the  slack 
side  of  the  rope  varies  with  the  weight  supported  by  the 
tension-carriage,  it  is  obvious  that  an  increase  of  this 
weight  will  increase  the  power  which  may  be  delivered  by 
the  rope.  As,  however,  the  horse-power  is  proportional  to 
the  difference  in  stress  in  the  driving  and  slack  sides  of 
the  rope,  the  less  weight  on  the  tightener  consistent  with 
obtaining  sufficient  frictional  resistance  to  slipping,  the 
better  will  the  ropes  work. 

An  example  of  rope-driving,  in  which  tension-carriages 


ROPE-DRIVING. 


33 


are  arranged  to  work  vertically,  is  shown  in  Fig.  17.  In 
this  case  the  engine  develops  about  45  horse-power  and 
runs  at  90  revolutions  per  minute,  corresponding  to  which 
the  velocity  of  rope  is  about  1700  feet  per  minute.  The 
fly-wheel  is  six  feet  in  diameter,  and  is  grooved  for  five  1£- 
inch  *opes»  The  main  sheave  on  the  jack-shaf '  is  also  6 


FIG.  17. — ROPE-DRIVE  WITH  VERTICAL  TENSION-CARRIAGES. 

feet  in  diameter,  and  is  grooved  for  five  ropes.  The  rope 
passes  continuously  from  the  fly-wheel  to  the  main  sheave, 
making  five  wraps;  then  over  the  deflecting-sheave  to  the 
horizontal  tension-carriage,  and  back  to  the  fly-wheel. 

The  jack-shaft  sheaves  are  grooved  for  four  1-inch  ropes, 
and  are  each  provided  with  a  friction-clutch,  giving  the 
line  two  shafts  and,  practically,  the  advantages  of  inde- 


34  ROPE-DRIVING. 

pendent  motors,  so  that  in  case  of  accident  to  either  the 
other  can  be  run  independently. 

The  sheaves  on  these  shafts  are  five  feet  in  diameter,  and 
are  grooved  for  five  1-inch  ropes, — four  wraps  being  used 
for  the  transmission  of  power;  the  other  groove  is  for  the 
return  from  the  vertical  tension-carriage,  which  is  shown 
just  beneath  the  pulley  on  the  line-shaft. 

Although  used  extensively  for  main  drives  and  in  many 
cases  for  intermediate  driving,  ropes  have  not  come  into 
use  for  general  work  throughout  the  factory.  In  certain 
cases  the  whole  of  the  driving  is  done  by  ropes,  no  cross- 
shafts,  gears,  nor  belts  being  used;  but  the  greater  conven- 
ience of  flat  belts  for  conveying  power  to  the  machine  in 
ordinary  shop-transmissions  is  so  thoroughly  acknowledged, 
that  any  attempt  to  substitute  rope  for  belting  in  general 
would  be  met  with  little  favor;  nor  would  this  be  practi- 
cable under  the  conditions  which  now  obtain  in  our  mills 
and  factories.  One  reason  for  this  is  the  difficulty  in 
shifting  a  rope  from  a  tight  to  a  loose  pulley. 

Numerous  devices  have  been  employed  for  this  purpose, 
and  in  some  few  cases  with  satisfactory  results;  but  the  sys- 
tem does  not  readily  lend  itself  to  such  work,  nor  would 
the  arrangements  that  have  been  adopted  be  generally  ap- 
plicable. Another  reason  for  its  non-employment  for 
general  work  is  the  size  of  pulleys,  and  distance  between 
shafts  which  is  necessary  for  satisfactory  working  with 
ropes,  and  which,  obviously,  would  exclude  its  use  in  many 
instances. 

Where  the  power  is  transmitted  to  a  shaft  whose  axis  is 
in  a  different  plane  from  that  of  the  driver,  it  is  often  in- 
convenient or  impracticable  to  use  bevel-gears  or  universal 
couplings.  For  such  special  cases  rope-driving  is  partic- 
ularly satisfactory,  as  a  flexible  rope  will  readily  conform 
to  any  direction,  and  transmit  its  full  power  when  arranged 
with  suitable  idlers. 


ROPE-DRIVING. 


35 


An  example  of  such  transmission  is  given  in  Fig.  18. 
This  drive  transmits  150  h.  p.  for  the  main  shaft  at  a  to 
another  at  b,  whose  axis  makes  an  acute  angle  with  the 
first,  and  which  is  several  feet  lower.  The  drive-sheave  a 
is  5  feet  3  inches  in  diameter  and  runs  at  160  revolutions 
per  minute;  it  is  grooved  for  six  IJ-inch  ropes,  which  have 
a  velocity  of  2800  feet  per  minute;  c  and  d  are  idlers,  the 
faces  of  which  are  nearly  parallel  to  the  drive-sheave;  e 


FIG.  18. — ROPE-DRIVE  WITH  SHAFTS  AT  AN  ANGLE. 

and  /  are  double  idlers,  there  being  two  sheaves  on  each 
shaft,  one  1|  inches  less  in  diameter  than  the  other.  This 
arrangement  of  idlers  in  quarter  twist  drives  of  this  class 
has  been  introduced  where  there  are  more  than  four  ropes 
in  the  system,  with  the  object  of  reducing  the  wear  conse- 
quent to  the  friction  produced  by  the  side  lead  of  the  rope. 
In  drives  where  there  are  more  than  eight  roes  a  cone 
has  been  used  to  a  limited  extent 

f  OF  THE 

UNIVERSITY^ 


36  BOPE-DRIVING. 

Such  a  practice  is,  however,  very  unsatisfactory,  as  the 
trouble  encountered  by  the  differential  driving  exerted  by 
each  rope  on  a  different  diameter  of  the  cone  is  greater 
than  that  which  it  attempts  to  obviate.  However,  by  mak- 
ing the  several  steps  on  the  cone  separate,  so  that  they 
form  a  series  of  independent  idlers  of  different  diameters,* 
the  difficulty  is  overcome. 

It  is  evident  that  the  employment  of  multiple  idle 
sheaves  of  equal  diameter  possesses  many  advantages  over 
a  multiple-grooved  pulley  when  used  as  guide-pulleys. 

In  this  case  each  sheave  is  independent  of  the  others, 
and  thus  prevents  in  a  large  measure  the  evils  due  to  dif- 
ferential driving  and  slip  which  would  otherwise  occur 
with  fluctuations  of  load. 

With  ordinary  transmissions  where  the  vertical  distance 
between  shafts  is  as  great  as  100  times  the  diameter  of  rope 
no  trouble  is  experienced  from  the  side  lead  of  the  rope, 
and,  usually,  no  provision  is  made  to  obviate  it. 

An  application  of  rope-driving  to  shafts  at  right 
angles,  embodying  several  excellent  features,  is  shown 
in  Fig.  19. 

The  plant  is  designed  to  transmit  250  horse-power  from 
a  14-ft.  fly-wheel,  a,  which  is  grooved  for  twelve  1-inch 
ropes.  The  line-shafts,  m  and  n,  are  driven  independently, 
and  each  drive  has  its  own  tension-carriage.  The  rope- 
sheave  1}  is  72  inches  in  diameter,  and  is  grooved  for  five 
ropes.  At  the  side  of  the  72-inch  sheave  is  a  single- 
grooved  idler,  i,  loose  on  the  shaft,  which  serves  as  a  guide 
for  the  rope  to  the  tension-carriage. 

The  substitution  of  a  loose  idler  for  an  extra  groove  on  the 
driven  pulley  in  rope  transmissions  is  due  to  Mr.  Spencer 
Miller,  although  it  has  long  been  in  use  in  cable-railway 

*  This  feature  is  the  subject  of  a  patent  granted  to  Mr.  Jghn 
Gregg,  March  11,  1890. 


ROPE-DRIVING. 


practice;  the  advantage  in  its  use  is  evident  when  we  con- 
sider that  the  tension-carriage,  drawing  out  the  stretch  of 
the  rope,  must  necessarily  drag  the  first  rope  through  the 


FIG.  19.— TRANSMISSION  AT  RIGHT  ANGLES. 

groove  of  the  pulley,  which  will  require  an  excessive  weight 
on  the  tightener  pulley  and  a  greater  length  of  time  before 


38  ROPE-DRIV1KG. 

equilibrium  is  restored.  By  having  this  groove  made  into 
an  individual  wheel  free  to  rotate  on  the  shaft,  this  diffi- 
culty is  overcome,  and  the  transmission  responds  very 
freely  to  changes  of  load,  so  that  when  heavy  machines 
are  thrown  on  and  off  the  ropes  are  not  set  in  vibration, 
but  the  tension-carriage  sheave  K  slides  back  and  forth 
on  the  track,  taking  up  the  shock,  with  a  minimum 
amount  of  wear  on  the  ropes. 

In  the  present  case  the  rope  runs  continuously  around 
the  fly-wheel  and  sheave  from  groove  to  groove.  As  it 
leaves  the  fly-wheel  at  the  left  hand  it  passes  over  the  idler 
*  to  the  tension-carriage  sheave  K,  which  is  suspended  on 
adjustable  hangers  from  a  single-pipe  track.  This  sheave 
is  tilted  by  means  of  the  adjustable  hangers,  so  that  the 
top  is  in  line  with  the  centre  of  the  groove  of  the  idler, 
and  the  bottom  is  in  line  with  the  centre  of  the  groove  of 
the  guide-sheave  j,  which  serves  to  carry  the  rope  back  to 
the  right-hand  groove  of  the  fly-wheel. 

The  employment  of  multiple  idlers  instead  of  a  multiple- 
grooved  idle  pulley  is  also  shown  in  this  figure,  where  power 
is  transmitted  from  the  shaft  n  to  another  o  at  right 
angles  to  the  first.  By  engaging  one  or  the  other  of  the 
driven  pulleys  e  or  f  by  means  of  a  clutch,  the  shaft  o  may 
be  driven  in  either  direction. 

More  recently  ropes  have  been  introduced  to  drive  dyn- 
amos and  special  isolated  machines,  and  where  the  dis- 
tance between  dynamos  and  engine  or  driving-shaft  is 
sufficiently  great  to  allow  a  moderate  sag  in  the  ropes,  such 
drives  have  been  found  to  work  very  satisfactorily,  provided 
other  conditions  are  favorable.  Where  the  distance  be- 
tween shafts  is  limited,  more  wraps  should  be  given  to  the 
rope  in  order  to  lessen  the  tension  in  each  member.  One 
great  fault  with  dynamo  drives  is  the  use  of  too  small  a 
pulley  on  the  armature-shaft.  We  can  point  to  a  score  of 
plants  using  rope  transmissions  from  a  jack-shaft  to  dyn- 


ROPE-DRIVIKG. 


39 


40  ROPE-DRIVING. 

amo  in  which  the  rope  is  overstrained  and  the  dynamo 
pulley  is  only  half  as  large  as  it  should  be,  in  consequence 
of  which  the  ropes  are  a  constant  source  of  trouble. 

With  cotton  ropes  the  pulley  may  be  somewhat  smaller 
than  that  used  for  a  similar  size  of  manilla  rope;  but  in 
any  case  there  is  a  certain  minimum  diameter  of  pulley 
which  should  be  used  for  any  given  rope.  (See  page  179.) 

Where  the  required  number  of  revolutions  cannot  be  at- 
tained with  the  size  of  pulley  imposed  by  the  various  con- 
ditions, if  the  jack-shaft  cannot  be  speeded  up  nor  a  larger 
driving-pulley  used,  in  such  cases  it  would  be  better  to  take 
out  the  rope  and  put  in  a  good  leather  belt. 

The  simplest  arrangement  of  rope  transmission  for  dyn- 
amos is  that  in  which  the  rope  is  carried  direct  from  the 
engine  fly-wheel  to  the  grooved  pulley  on  the  armature 
shaft,  as  shown  in  Fig.  20,  which  represents  the  system 
used  in  the  station  of  the  Liverpool  Overhead  Railway.* 

There  are  four  horizontal  compound  condensing  engines 
with  cylinders  15^  and  31  inches  in  diameter,  36  inches 
stroke,  each  of  which  is  connected  to  a  separate  generator 
by  means  of  19  ropes  1J  inches  diameter.  Each  engine  is 
rated  at  400  h.  p.  when  running  at  100  revolutions  per 
minute  with  120  pounds  initial  pressure;  as  the  fly-wheels 
are  14  feet  in  diameter,  the  rope  velocity  will  thus  be  about 
4400  feet  per  minute. 

In  general  it  is  more  desirable  to  drive  the  machines 
through  an  intermediate  jack-shaft,  especially  so  in  those 
cases  where  a  varying  amount  of  current  is  required,  as, 
for  instance,  in  the  lighting  of  public  buildings.  Such  an 
arrangement,  when  the  jack-shaft  is  provided  with  suitable 
friction  or  jaw  clutches,  will  permit  machines  to  be  thrown 
on  or  off  as  desired.  The  use  of  an  intermediate  shaft  also 
permits  the  attainment  of  the  requisite  speed  of  the  dyn- 

*  Power,  May,  1893. 


ROPE-DRIVING.  41 

amo  with  moderate  proportions  of  pulleys.  In  fact  with 
many  of  the  smaller  engines  in  use  a  jack-shaft  is  essential 
if  we  wish  to  use  rope-driving. 

Take,  .for  example,  a  75-h.-p.  Corliss  engine,  running  at 
85  revolutions  per  minute;  the  diameter  of  fly-wheel  for 
this  engine  is  10  feet,  and  if  we  wish  to  drive  the  dynamo 
direct  through  a  f-inch  rope,  the  pulley  on  the  armature- 
shaft  should  be,  preferably,  not  less  than  24  inches  diam- 
eter; the  speed  of  the  armature  would  then  be  only  425 
revolutions  per  minute.  To  obtain  a  suitable  speed,  the 
driven  pulley  could  not  be  more  than  about  12  inches  in 
diameter,  and  with  larger  ropes  this  difference  would  be 
still  more  pronounced. 

A  similarly  rated  high-speed  engine  runs  at  230  revolu- 
tions per  minute,  and  is  provided  with  a  fly-wheel  5  feet  in 
diameter;  with  the  same  24-inch  pulley  on  the  d}mamo  the 
speed  of  the  latter  when  driven  direct  would  be  not  more 
than  570  revolutions  per  minute,  so  that  in  this  case  also 
the  driven  pulley  would  have  to  be  reduced  very  much 
below  that  size  which  has  been  found  best  adapted  to  the 
work. 

It  is  true  that  the  diameter  of  driving-wheel  on  the  en- 
gine-shaft could  be  increased,  and  this  is  sometimes  done. 
In  the  cases  quoted  the  diameters  necessary  to  give  the  re- 
quired speed  would  be  about  20  feet  for  the  Corliss  and  7 
feet  for  the  automatic  engine.  As  these  sizes  give  a  cir- 
cumferential velocity  within  the  limit  of  safety  from  the 
action  of  centrifugal  force  of  the  metal  in  the  rim,  it  would 
be  highly  desirable  to  use  such  driving-wheels  if  other  prac- 
tical considerations  did  not  preclude  their  use.  A  driving- 
wheel  7  feet  in  diameter  could  readily  be  used  on  the  high- 
speed engine  without  materially  augmenting  the  journal 
friction,  and  the  increased  rim  speed  would  be  a  beneficial 
factor  in  preventing  momentary  fluctuations  due  to  change 
in  load. 


42  ROPE-DRtVlKG. 

With  the  Corliss  engine,  however,  the  increased  weight 
due  to  a  large  built-np  fly-wheel  20  feet  in  diameter  would 
usually  debar  its  use  on  an  engine  of  this  capacity;  in  ad- 
dition to  this  the  large  diameter  would  prevent  its  use  in 
many  locations,  even  if  the  increased  weight  and  loss  of 
power  were  no  hindrance.  Under  these  conditions  the  use 
of  a  jack-shaft  is  the  most  suitable  arrangement. 

In  many  cases  it  is  desirable  to  use  two  engines  so  ar- 
ranged that  either  or  both  may  furnish  the  power  to  any 
one  of  several  dynamos,  as  shown  in  Fig.  21,  which  repre- 
sents a  175-h.-p.  rope-transmission  plant  erected  by  the 
Link-belt  Engineering  Company  in  the  Virginia  Hotel, 
Chicago,  where  two  Corliss  engines  are  each  connected  to 
a  jack-shaft,  having  five  counter-drives  to  the  dynamos. 
Both  the  driven  and  driving  sheaves  on  the  jack-shaft  are 
loose  on  the  shaft,  and  are  connected  to  it  by  means  of  fric- 
tion or  jaw  clutches,  thus  permitting  either  or  both  engines 
to  be  run,  or  any  one  of  the  five  dynamos  to  be  thrown  in  or 
out  of  use.  The  positive  jaw-clutch  is  used  on  the  driven 
sheave,  as  it  does  not  readily  get  out  of  order  and  is 
preferred  by  many  engineers  to  the  average  friction- 
clutch,  especially  in  those  cases  where  much  power  is 
transmitted. 

If  it  is  desired  to  couple  one  engine  to  the  shaft  while 
the  other  is  running,  the  former  is  speeded  up  until  the 
loose  driven  sheave  comes  up  to  the  speed  of  the  shaft, 
when  the  dog-clutch  may  be  readily  thrown  in  gear  with- 
out shock.  With  this  arrangement  there  is  a  strong  ten- 
dency for  the  bearings  of  the  driven  sheaves  to  heat  when 
not  coupled  to  the  shaft;  for  this  reason  provision  should 
be  made  to  reduce  the  friction  between  the  loose  pulley  and 
the  shaft  by  relieving  the  tension  on  the  rope  when  not  in 
use,  or,  what  is  much  better,  the  loose  sheave  should  be 
mounted  upon  a  hollow  sleeve  supported  in  pedestals  inde- 
pendently of  the  shaft,  as  noted  in  description  of  plant 


ROPE-DRIVIXG. 


43 


shown  in  Fig.  10;  see  also  Fig.   33.     In  this  way  either 
pulley  may  be  at  rest  and  impose  no  friction  on  the  shaft. 


FIG.  21. — ROPE  TRANSMISSION  WITH  JACK-SHAFT. 

If  Tl  is  the  maximum  stress  in  a  rope,  T9  the  stress  in 
the  slack  part,  and  P(=Tl  —  Tt)  is  the  driving  force,  then, 


44 


ROPE-DRTVIXG. 


as  we  shall  show  subsequently,  the  ratio  of  the  maximum 

T 

stress   Tl  to  the  driving  force  P,  or  -~,  will  vary  from 

about "1.5  to  5.5 — depending  upon  the  speed  of  rope,  the 
coefficient  of  friction,  and  the  angle  embraced  by  the  rope 
on  the  circumference  of  the  pulley.  In  order,  then,  to 
have  the  transmitting  force  P  as  large  as  possible  for  a 
maximum  tension  Tl9  the  tension  in  the  slack  part  of  the 
rope  necessary  for  adhesion  must  be  reduced  to  a  mini- 
mum. To  a  certain  extent  this  can  be  obtained  by  de- 
creasing the  angle  between  the  sides  of  the  groove,  but  if 
carried  too  far  this  is  a  detriment  rather  than  an  advan- 
tage, for  if  the  angle  is  sufficiently  acute  the  rope  will 
wedge  and  require  more  or  less  force  to  pull  it  out  of  the 
groove.  The  remaining  expedient  is  to  increase  the  arc  of 
contact. 

It  can  be  shown  that  the  friction  of  a  cord  or  rope 
wrapped  upon  a  fixed  cylinder  is  independent  of  the  di- 
ameter of  the  cylinder,  and  that  it  increases  very  rapidly 
with  an  increased  arc  of  contact.*  If  the  conditions  are 
such  that  the  coefficient  of  friction  0  =  one  third,  a  ten- 
sion of  one  pound  at  the  end  T9  of  the  rope,  Fig.  22,  will 
support  a  strain  at  the  end  Tl  of: 


1.69  pounds  foi 

.-.   2.85 

8.12 

65.94 

530.43 

4,348.53 


an  arc  of  contact  equal  to  £  coil. 


Therefore  by  increasing  the  number  of  wraps  around  the 
cylinder  it  is  possible  to  increase  the  difference  between  the 
tensions  in  each  part  of  the  rope  almost  indefinitely.  It 
will  be  noticed  here  that  the  rope  is  not  in  flying  motion, 


*Weisbach,  vol.  i.  p.  360. 


ROPE-DRIVING.  45 

which  would  cause  an  equal  centrifugal  force  to  be  set  up 
in  each  member,  thus  altering  the  ratio  of  stress.  As  the 
centrifugal  force  varies  with  the  square  of  the  velocity, 
there  is  with  an  increasing  speed  of  the  rope  a  decreasing 
useful  force  and  an  increasing  total  tension  01?  the  slack 
side;  but  up  to  a  given  limit,  which  we  shall  subsequently 
show  lies  between  4000  and  6000  feet  per  minute,  the 
total  power  transmitted  for  a  given  maximum  tension  in 
t-he  rope  will  increase  with  the  velocity. 

The  great  advantage  thus  obtained  by  increasing  the  ad- 
hesion was  very  early  applied  to  numerous  mechanical  de- 
vices, chiefly  for  winding  and  hoisting  purposes,  and  later 
for  haulage  systems.  We  are  indebted  to  Willis*  for  the 
following  quaint  account  and  sketch  (Fig.  23)  of  an  ar- 
rangement for  obtaining  a  continuous  motion  with  a  con- 
tinuous travelling-coil,  first  suggested  by  the  author  of  the 
article,  Sir  Christopher  Wren,  over  two  hundred  years  ago. 

"A    DESCRIPTION   AND   SCHEME    OF    DK.   WREN'S    INSTRU- 
MENT  FOR   DRAWING    UP   GREAT   WEIGHTS 
FROM   DEEP   PLACES." 
Read  May  5,  16~0. 

"  Having  considered,  that  the  ways  hitherto  used  in  all 
Engina  for  winding  up  Weights  by  Eoaps  have  been  but 
two,  viz.  the  fixing  one  end  of  a  roap  upon  a  cylinder  or 
Barril,  and  so  winding  up  the  whole  coyle  of  roap  ;  the 
other  by  having  a  chain  or  a  loose  roap  catching  on  teeth, 
as  is  usual  in  clocks:  but  finding  withall  that  both  these 
wayes  were  inconvenient  the  first,  because  of  the  riding  of 
much  roap  in  winding  one  turn  upon  another;  the  other, 
because  of  the  wearing  out  of  the  chain  or  roap  upon  the 
teeth,  I  have,  to  prevent  both  these  inconveniences,  devised 
another  to  make  the  weight  and  its  counterpoyse  bind  on 

*  "  Principles  of  Mechanism,"  p.  439  et  seq. 


46 


ROPE-DRIVING. 


the  cylinder,  which  it  will  doe  if  it  be  wound  three 
about. 

"But  because  it  will  then  in  turning,  scrue  on  like  a 
worm,  and  will  need  a  Cylinder  of  a  very  great  length, 
therefore  if  there  be  two  cylinders  each  turned  with  three 
notches  and  the  notches  be  placed  alternately,  the  convex 


FIG.  22. 


FIG.  24. 


FIG.  23. 


edges  to  the  concave  as  in  the  figure  here  adjoyned,  the 
roap  being  wound  three  times  about  both  cylinders,  will 
bind  firmly  without  slyding  and  work  up  the  weight  with 
a  proportionable  counterpoyse  at  the  other  end  of  the 
Koap." 

The  method  of  obtaining  increased  adhesion  for  a  given 
wrap  by  increasing  the  number  of  coils  in  contact  with 


ROPE-DRIVING. 


47 


each  pulley  has  long  been  in  use  in  rope-driving.  In  some 
of  the  earlier  applications  the  grooves  of  the  pulleys  were 
semicircular  in  section,  and  of  sufficient  size  to  allow  the 
rope  to  embrace  the  entire  circumference  of  each  pulley  as 
represented  in  Fig.  24.* 

A  better  arrangement  is  that  shown  by  Over  man,  f  in 
which  the  advantage  of  the  angular  groove  is  obtained, 
and  the  ropes  are  not  worn  by  rubbing  against  each  other. 
This  is  shown  in  Fig.  25,  and  is  thus  described: 


FIG.  25. 

"If  the  pulley  A  is  grooved,  of  which  at  least  two  are 
fastened  to  the  same  shaft,  the  rope  is  directed  on  one  of 
these  pulleys,  and  passing  around  it  goes  to  J3,  which  re- 
volves on  an  inclined  axis,  such  that  the  rope  will  be  re- 
ceived from  A'  and  delivered  to  A  in  the  plane  of  the 
grooves.  The  number  of  pulleys  may  be  multiplied  to 
gain  adhesion.  This  method  of  augmenting  friction  is 
preferable  to  the  tension-roller,  as  no  increase  of  tension 
is  required;  and  it  has  the  additional  advantage  of  bend- 
ing the  rope  in  the  same  direction,  which  makes  it  more 
durable." 

*  Willis.     Principles  of  Mechanism. 
|  Overman's  "  Mechanics,"  1851. 


48 


ROPE-DRIVING. 


A  similar  arrangement  was  introduced  in  the  San  Fran- 
cisco Cable  Railway  in  1877.  In  this  case,  Fig.  26,  the 
endless  hauling-cable  is  passed  alternately  backward  and 
forward  over  two  grooved  drums  a  sufficient  number  of 
times  to  obtain  the  necessary  driving  adhesion. 

Positive  motion  is  imparted  to  the  drum  A,  which  hauls 
in  the  cable,  while  a  second  pulley,  B9  acting  as  an  idler, 


FIG.  26.— COIL-FRICTION,  STREET-RAILWAY  CABLE. 

increases  the  f notional  grip  of  the  cable  on  the  drum,  by 
virtue  of  the  increased  arc  of  contact  due  to  the  number 
of  wraps.* 

When,  however,  the  cable  permanently  lengthens  by 
stretching,  the  drum  B  may  be  moved  further  back  by 
means  of  the  sliding-base  0  so  as  to  take  up  the  resulting 
slack. 

By  the  use  of  this  winder  pulley  the  property  of  fric- 
tional  adhesion  produced  by  successive  coiling  is  perfectly 

*"  Cable  or  Rope  Traction,"  J.  Bucknall  Smith,  C.  E.;  Engi- 
neering, London,  1887. 


effectual,  for,  although  each  coil  is  only  in  contact  with  a 
semi-circumference,  the  accumulation  of  frictional  resist- 
ance is  produced  precisely  as  if  entire  circumferential 
grooves  were  employed. 

However  desirable  such  winder  pulleys  may  be  for  cable 
haulage  or  hoisting  purposes,  their  advantage  is  greatly 
overestimated  when  applied  to  continuous  rope-transmis- 
sion. A  little  consideration  will  show  that  the  frictional 
adhesion  produced  by  a  tension  weight  acting  on  a  running 
rope  with  numerous  wraps  is  entirely  different  from  coil- 
friction.  In  the  latter  we  have  an  accumulation  of  friction 
by  which  a  small  resistance  applied  at  one  end  of  the 
rope  is  able  to  hold  an  enormously  greater  load  at  the  other 
end. 

In  the  continuous-rope  system  of  power  transmission, 
however,  the  load  is  distributed  among  all  the  wraps,  so 
that  when  properly  adjusted  each  wrap  carries  an  equal 
proportion  of  the  load  and  is  subjected  to  an  equal  resist- 
ance on  its  slack  side.  There  are  few  cases  where  the  com- 
bination of  the  winder-pulley  with  the  continuous- rope 
system  offers  any  decided  advantage  over  other  methods. 

There  is  an  incidental  advantage  in  using  a  winder, 
especially  in  those  cases  where  the  difference  in  diameters 
between  the  driver  and  follower  is  quite  appreciable;  under 
such  conditions  the  adhesion  of  the  rope  on  each  pulley 
may  be  made  more  nearly  uniform  by  the  employment  of  a 
winder,  and  there  is  less  liability  of  the  rope  slipping  in  its 
groove,  but  this  may  usually  be  obtained  more  satisfactorily 
by  other  means  (see  page  168). 

In  numerous  cases  ropes  using  winder-pulleys  have  been 
installed  without  regard  to  the  work  to  be  done  or  strain 
put  upon  the  ropes,  and  many  of  the  evils  of  rope-driving 
are  directly  traceable  to  this  cause. 

Many  engineers  ar,e  opposed  to  using  the  winder-pulley 


50  HOPE-DRIVING. 

in  any  form  whatever,  but  occasionally  it  may  be  used  to 
advantage. 

In  outdoor  or  long-distance  transmissions,  or  in  special 
cases  where  it  is  desired  to  transmit  a  maximum  power 
without  undue  stress  in  the  rope,  or  in  particular  cases 
where  the  ordinary  working  stress  may  be  exceeded,  a 
winder-pulley  may  frequently  be  used  to  advantage,  if  the 
slack-side  tension  be  reduced  accordingly.  Using  a  winder- 
pulley  and  increasing  the  back  tension  will  permit  a  very 
large  increase  in  the  power  transmitted;  but  since  this  im- 
poses an  excessive  strain  on  the  rope,  it  soon  wears  out,  and 
is  a  constant  source  of  trouble. 

The  gain  in  power  by  increasing  the  adhesion  will  be  at 
the  expense  of  journal-friction,  which  is  thus  augmented 
by  the  employment  of  wider-faced  driving  and  driven  pul- 
leys, in  addition  to  that  due  to  one  or  two  more  winder- 
pulleys;  the  wear  of  the  rope,  both  external  and  internal^ 
will  also  be  greatly  increased  on  account  of  the  greater 
number  of  flexures  given  to  the  rope  in  passing  over  the 
winder-pulleys. 

The  use  of  a  winder-pulley  at  each  end  of  a  long  drive 
in  which  only  a  single  strand  runs  from  the  driver  to  driven 
pulley  is  an  example  of  the  application  of  coil-friction  to 
the  continuous-rope  system;  in  this  case  both  the  working- 
load  and  the  back  tension  is  carried  by  one  rope  instead  of 
being  distributed  among  several  wraps,  as  usually  happens 
in  this  system. 

That  the  percentage  of  gain  is  not  as  great  as  might  be 
expected  from  the  employment  of  coil-friction,  will  be  seen 
from  the  following  considerations : 

The  ratio  of  tensions  in  the  tight  and  slack  sides  of  a 
rope  running  over  two  pulleys  is  dependent  upon  several 
factors,  and  may  be  determined  from 

a*.  = 


ROPE-DRIVING.  51 

in  which  T^  =  tension  in  tight  side  of  rope, 
jP2  =  tension  in  slack  side  of  rope, 
e    =  base  of  hyp.  log  =  2.7183, 
0   =  coefficient  of  friction, 
a   =  arc  of  contact  (circular  measure). 

In  this  case  the  influence  of  centrifugal  force  is  neg- 
lected. 

Since  the  power  transmitted  by  a  wrapping  connector 
is  dependent  upon  the  difference  of  tensions  in  the  tight 
and  slack  sides,  it  is  evident  that  with  an  assumed  total 
tension  Tt  the  available  force  for  transmitting  power  will 
increase  as  Tz  decreases. 

But  T9  may  diminish  as  e^a  increases,  that  is,  since  e 
is  a  constant  and  0  is  constant  for  a  given  pulley  and  ma- 
terial, T9  decreases  as  a  increases;  hence  if  we  increase 
the  arc  of  contact  the  tension  in  the  slack  side  of  the  pul- 
ley may  be  decreased  in  a  ratio  greater  than  unity,  depend- 
ing upon  the  factors  involved;  in  which  case  the  net  force 
P  available  for  transmission  will  be  increased,  while  the 
original  assumed  allowable  tension  remains  the  same. 

For  example,  if  Tl  —  200  pounds,  0  =  0.3,  a  =  2.88 
(a°  =  165°),  we  shall  have 

900 

.,uu     _  _ 

•*!  /O  ryi  Q\0.3X2.88          ua:> 


and  the  net  force  P  =  200  —  84  =  116  pounds. 

Under  the  same  conditions,  if  we  pass  the  rope  from  the 
driver  over  a  winder-pulley  back  and  forth  twice,  and  then 
to  the  driven  pulley  and  its  winder  in  the  same  way,  we 
should  be  able  to  transmit  a  little  more  than  one  and  a  half 
times  as  much  power  at  the  same  speed  without  increasing 
the  working  tension  in  the  rope. 

T 

In  this  case  a  —  12.56  and  ~  —  43.1;  hence 

•*•  3 


52  HOPE-DRIVING. 

ra  =  4.6  and  P  =  200  -  4.6  =  195  +  pounds. 

Horse-power  in  second  case  _  195  _ 
*  Horse-power  in  first  case          116  "~ 

Now,  if  it  were  practicable  to  maintain  the  same  slack- 
side  tension  T^  in  these  two  instances  and  increase  the 
driving-side  tension  under  the  conditions  of  the  second 

T 
case,  viz.  -^  =  43.1,  we  should  have 

P=Tl-T9  =  (43.1  X  84)  -  84  =  3536  pounds, 

3536 

and  the  ratio  of  power  transmitted  will  now  be  or 

116 

thirty  times  as  great  as  before. 

These  results  indicate  that  while  we  may  vastly  increase 
the  driving-side  tension  for  a  given  slack-side  tension,  by 
using  a  winder  in  the  manner  indicated,  that  is,  with  a 
single  wrap  connecting  driver  and  follower,  yet  if  we  wish 
to  maintain  an  assumed  maximum  working  tension  for  a 
given-sized  rope  the  percentage  of  gain  will  not  be  very 
great  under  the  usual  requirements  of  rope-driving. 

For  a  temporary  drive  the  working  strain  may  be  in- 
creased to  about  twice  the  usual  value,  but  for  a  permanent 
installation  the  usual  working  value  should  not  be  ex- 
ceeded. 

Where  a  number  of  ropes  are  employed  on  a  short-drive 
it  is  questionable  whether  the  winder-pulley  possesses  suffi- 
cient advantages  to  warrant  its  employment  in  place  of  the 
continuous-wrap  or  individual-rope  systems.  In  any  case 
the  conditions  should  be  carefully  considered,  and  the  ac- 
tual gain  compared  with  the  various  losses  involved. 

A  recent  example  illustrating  the  application  of  the 
winder-pulley  is  shown  in  Fig.  27,  which  represents  the 
system  of  rope-driving  installed  by  Messrs.  Hoadley  Bros, 
in  the  Fifty-second  Street  electric  power-house  of  the 
Chicago  City  Railway  Company. 


54  ROPE-DBIVItfG. 

The  plant  is  designed  for  10  generators  of  the  Westing- 
house  No,  6  type,  running  300  revolutions  per  minute. 
There  are  also  to  be  ten  24-inch  by  48-inch  engines  of  the 
improved  Wheelock  type,  arranged  in  five  pairs,  two  oi 
which  are  now  in  operation.  These  run  at  100  revolutions 
per  minute  with  100  pounds  boiler-pressure.  The  power 
usually  varies  from  about  200  to  1000  h.  p.,  but  [during  the 
heavy  traffic  throughout  the  summer  each  pair  of  engines 
has  frequently  transmitted  1500  h.  p. 

These  engines  have  a  built  up  fly-wheel  (10  segments)  18 
feet  in  diameter,  39  inches  face,  which  weighs  about 
50,000  pounds.  The  rim  is  grooved  for  21  wraps  of  1£- 
inch  manilla  rope.  The  driven  pulleys  are  6  feet  in 
diameter,  and  contain  32  grooves  for  the  rope,  which  runs 
about  5600  feet  per  minute.  Between  the  driven  pulleys 
and  the  engine  fly-wheel  there  is  placed  a  6-foot  winder, 
containing  11  grooves,  around  which  the  rope  is  car- 
ried before  passing  to  the  tension-sheave  (Fig.  28),  which 
in  the  present  arrangement  is  placed  horizontally  above 
the  engine  near  the  ceiling,  as  shown  in  Fig.  29.  Thus  the 
rope  is  wound  around  the  engine  fly-wheel  and  the  driven 
pulley,  making  20  wraps;  then  it  is  carried  from  the 
driven  pulley  to  the  winder  back  and  forth  11  times, 
thence  it  is  led  over  vertical  guide-pulleys,  7  feet  in  diam- 
eter, to  the  horizontal  tension-sheave  54  inches  in  diameter, 
then  down  over  another  vertical  guide-pulley  to  the  fly- 
wheel, where  it  started.  By  this  means  the  arc  of  contact 
of  each  member  of  the  driving-rope  is  increased  practically 
180  degrees  when  all  the  ropes  have  adjusted  themselves 
to  the  load,  so  that  the  power  transmitted  with  the  same 
tension  in  the  rope  will  be  about  forty  per  cent  more,  if  we 
neglect  friction,  than  would  be  transmitted  by  the  twenty- 
one  wraps  over  the  fly-wheel  without  the  use  of  a  winder. 

The  net  gain  will  be  considerably  less,  owing  to  the  vari- 
ous losses  which  this  system  entails. 


55 


56 

The  application  of  ropes  to  transmit  the  power  from  a 
water-wheel  to  a  line-shaft  40  or  50  feet  or  more  above  the 
axis  of  the  wheel  has  lately  received  considerable  attention, 
and  offers  many  advantages  over  the  ordinary  method  in 
which  a  vertical  shaft  is  used.  The  extreme  weight  of  the 
latter  in  many  cases  makes  it  a  difficult  matter  to  provide 
a  suitable  bearing  to  support  it.  In  such  cases  a  horizon- 
tal turbine  is  used,  and  the  wheel-shaft  carrying  the  rope- 
pulley  is  extended  and  suitably  supported,  as  shown  in 
Fig.  30. 

The  station  of  the  Brush  Electric  Light  and  Power  Co. 
at  Niagara  Falls  is  driven  in  this  way. 

A  line-shaft  runs,  through  the  building,  with  one  end  ex- 
tending over  the  wheel-pit;  to  this  are  belted  the  gener- 
ators in  the  usual  manner.  Seventy-five  feet  below  this 
shaft  is  located  a  15-inch  horizontal  Victor  turbine  in  a 
case  of  boiler-iron,  its  shaft  extending  to  bearings  supported 
by  bridge-trees,  which  in  turn  are  carried  by  the  foundation 
I  beams  that  support  the  wheel-case.  This  shaft  carries  an 
iron  pulley  40  inches  in  diameter,  grooved  for  12  f-inch 
manila  ropes.  (Cotton  was  tried,  but  was  not  satisfactory 
here.) 

The  pulley  on  the  driven  shaft  above  is  of  wood,  70 
inches  in  diameter.  The  driving  side  of  the  ropes  hang- 
perpendicularly,  and  are  free  from  the  driver  to  the  driven 
pulley. 

The  slack  side  has  two  idlers  or  guide-pulleys,  one  of 
which  is  situated  immediately  below  the  driven  pulley,  and 
the  other  is  about  20  feet  above  the  driver.  A  tightener  is 
adjusted  in  a  running  frame,  in  line  with  the  driven  and 
upper  guide-pulleys. 

In  putting  on  the  rope  the  following  course  is  taken : 
"  Commencing  at  one  side  of  the  pulleys  the  rope  is  passed 
around  from  the  driver  to  the  driven  pulley  in  every  alter- 
nate groove  until  the  opposite  side  is  reached,  thence 


around  the  tightener-pulley  in  the  running  frame,  which  is 
hung  on  an  incline  in  such  a  manner  that  its  discharg- 
ing side  is  in  line  with  the  side  of  the  driver  pulley 
whence  we  started .  The  remaining  grooves  are  then  filled, 
and  the  ends  of  the  rope  are  spliced  in  position  around  the 
idler.  Thus  it  is  readily  seen  that  there  are  two  strands  of 
the  rope  on  the  idler  at  all  times.  The  object  of  this  is  to 
have  one  solid  piece  of  rope  on  the  idler  at  the  same  time 
that  the  splice  is,  so  as  to  relieve  the  spliced  piece  of  the 
strain  of  the  idler.  This  system  is  giving  far  better  satis- 
faction than  has  the  upright  shaft  to  those  who  have  tried 
both."* 

The  plant  designed  by  Mr.  Robert  Cartwright  for  the 
electric  station  of  the  Citizens'  Light  and  Power  Company 
of  Rochester,  N.  Y.,  is  worthy  of  careful  study,  and  may 
be  considered  a  representative  modern  plant,  adapted  to 
use  steam  or  water-power,  and  employing  both  ropes  and 
belting,  f 

Fig.  31  represents  a  cross-section  of  the  station,  and 
shows  the  general  arrangement  of  the  plant.  The  water- 
wheels  are  twin  Poole-Leffel  central-discharge  turbines, 
23  inches  in  diameter,  and  at  a  speed  of  560  revolutions 
per  minute,  under  a  head  of  92  feet  6  inches,  develop  500 
horse-power  each,  with  a  discharge  of  3800  cubic  feet  of 
water  per  minute.  The  wheels  proper  are  made  of  phos- 
phor-bronze, with  buckets  of  Otis  steel,  tinned.  The  wheel 
bed-plates  are  heavy  cast-iron  box  sections,  machined  and 
bolted  together  with  heavy  bolts  fitting  reamed  holes.  The 
wheel-shaft  is  4f  inches  diameter,  running  in  adjustable 
babbitt-lined  bearings.  A  rope-wheel  4  feet  in  diameter  is 
keyed  on  the  shaft,  and  is  grooved  for  fifteen  1^-inch 
manilla  "Stevedore  "  ropes,  made  with  four  strands  and  a 
core,  worked  in  with  plumbago  in  the  process  of  making. 

*  F.  E.  Pritchard,  Elect.  World,  April  16,  1892. 
f  See  Trans.  Am.  Soc.  C.  E.,  vol.  xxx.,  1894, 


FIG.  31.— SECTION  THROUGH  POWER  STATION, 


59 


From  the  4-ft.  wheel  15  ropes  run  to  a  rope-wheel  on  the 
line-shaft  above,  76.8  inches  in  diameter,  and  grooved  for 
sixteen  1^-inch  ropes.  The  ropes  being  endless,  the  idler- 
strand  is  passed  over  a  5-ft.  single-grooved  wheel,  placed 
in  a  movable  frame.  The  frame  traverses  in  iron  guides 
and  maintains  by  its  weight  a  constant  tension  on  all  the 
ropes.  This  is  made  adjustable  for  the  amount  of  tension 


FIG.  32. — PLAN  OF  POWER  STATION. 

by  the  application  of  counter-weights  to  the  frame.  The 
speed  of  the  line-shafts  is  350  revolutions  per  minute,  and 
the  rope  travel  is  7037  feet  per  minute.  The  water-wheels 
are  supplied  from  a  steel  flume  7  feet  in  diameter.  From 
the  horizontal  portion  of  the  flume  a  4-ft.  pipe  leads  down 
to  each  wheel,  and  has  a  geared  48-inch  Chapman  valve  at 
the  lower  end,  between  pipe  and  penstock,  as  shown  in 


60 


ROPE-DRIVIKG. 


ROPE-DRIVING.  61 

Fig.  31.  These  valves  are  fitted  with  a  12-inch  by-pass, 
for  the  purpose  of  equalizing  the  pressure  on  both  sides  of 
the  large  valve  in  opening  or  closing. 

A  horizontal  "  Woodbury  "  compound  condensing  slide- 
valve  engine,  with  extra  heavy  bed-plate,  is  set  in  the 
power-room  at  point  marked  in  Fig.  32  "  Engine  No.  I." 
Steam-cylinders  are  placed  with  the  large  cylinder  out- 
side, so  that  pistons  and  rod  may  be  easily  removed.  Cylin- 
ders are  19  inches  and  31  by  24  inch  stroke.  At  167  revolu- 
tions, with  a  boiler-pressure  of  110  pounds  per  square  inch, 
vacuum  22  to  24  inches  and  cutting  off  at  T5¥  stroke,  the 
engine  is  rated  at  500  horse-power,  and  is  guaranteed  to 
produce  a  horse-power  on  an  evaporation  of  20  pounds  of 
water  per  hour.  The  crank-shaft  is  a  steel  forging  in  one 
piece.  Journals  are  11^  inches  diameter  by  21  inches 
long.  Crank-pin  11^  inches  diameter  by  8£  inches  long. 
The  end  carrying  the  rope-driving  wheel  has  an  outboard 
bearing.  Governor  balance-wheel  is  8^  feet  diameter  by 
25-inch  face.  Rope-driving  wheel  is  cast  in  halves  10  feet 
6  inches  diameter,  and  grooved  for  fifteen  If-inch  ropes, 
These  ropes  lead  to  a  5-ft.  rope-wheel  on  the  line-shaft 
above,  with  same  arrangement  for  tightener  as  is  applied 
to  the  water-wheels.  Rope  speed  of  engine-drive  is  5500 
feet  per  minute. 

The  line-shafts  are  of  hammered  iron  5  inches  in  diameter, 
and  arranged  with  heavy  floor  pedestals,  fitted  with  self- 
adjusting,  ring-oiled,  babbitt-lined  bearings.  The  rope- 
wheels  are  placed  on  heavy  cast-iron  quills,  furnished  with 
Hill  friction-clutches  of  500-horse  power  capacity  each. 
By  a  series  of  jaw-clutches,  pulleys,  and  belts  any  line- 
shaft  can  be  operated  from  any  water-wheel  or  engine,  all 
the  line-shafts  making  the  same  number  of  revolutions. 
Fig.  33  shows  in  detail  the  quill,  clutch,  and  bearings. 


62  ROPE-DRIVING. 


CHAPTER  IV. 

use  of  ropes  in  connection  with  portable  tools  and 
travelling-cranes  has  long  been  established,  and  their  con- 
venience and  adaptability  to  a  wide  range  of  work  make 
them  a  necessity  in  many  shops.  The  advent  of  the  small 
electric  motor  in  our  machine-shops  will,  however,  proba- 
bly replace  to  a  large  extent  all  other  forms  of  special 
transmission  for  portable  tools,  as  it  is  already  replacing 
countershafts  and  belting  for  machine-driving  in  many 
cases. 

One  of  the  greatest  fields  of  usefulness  for  rope-driving 
is  in  the  transmission  of  power  to  a  moderate  distance, 
under  conditions  which  are  unfavorable  to  the  use  of  belts 
or  shafting. 

With  rope-driving  one  is  enabled  at  a  comparatively 
small  cost  to  transmit  power  in  any  direction  to  a  building 
remotely  situated  from  the  source  of  power,  which  would 
otherwise  require  a  long  and  expensive  line-shaft  or  an  in- 
dependent engine  or  other  motor.  The  facility  with  which 
it  may  be  carried  in  any  direction  across  rivers,  canals,  and 
streets,  above  or  under  ground,  up  hill  and  down,  over 
houses  and  into  buildings,  is  a  feature  very  favorable  to 
the  further  extension  of  rope  transmission;  but  the  rapid 
progress  which  has  been  made  in  the  development  of  elec- 
trical transmission  has  limited  the  economical  application 
of  ropes  to  moderate  distances.  There  are,  however,  cer- 
tain limits  between  which  the  transmission  of  power  by 
ropes  is  yet  more  efficient  than  by  any  other  known  method. 

The  employment  of  ropes  for  this  purpose — i.e.,  trans- 
mission of  power  to  a  distance — is  not  a  recent  application. 


ROPE-DKIVING.  63 

The  first  method  of  transmission  of  power  to  any  consider- 
able distance  was  made  in  1850  by  C.  F.  Hirn  at  Logel- 
bach,  near  Colmar,  Alsace.*  "  The  works  consisted  of  a 
large  number  of  buildings  separated  at  some  distance  from 
one  another,  which  were  required  to  be  changed  into  a 
weaving  factory.  As  there  was  but  one  steam-engine  on 
the  works,  the  expense  of  transmitting  power  to  the  various 
buildings  by  ordinary  shafting  (the  shortest  length  of  which 
was  84  yards),  or  of  erecting  separate  engines,  would  neces- 
sarily have  been  great;  and  the  desire  to  obviate  this  ex- 
pense resulted  in  the  adoption  of  the  telodynamic  system. 

The  first  plan  adopted  was  the  use  of  a  band  of  steel  172 
yards  long,  ^  inch  thick,  and  2  inches  broad.  This  was 
slung  as  an  endless  band  over  two  wooden  rollers  or 
pulleys,  6  feet  6  inches  diameter,  which  were  placed  84 
yards  apart  and  made  120  revolutions  per  minute,  giving  a 
speed  of  28  miles  an  hour  in  the  band.  In  practice  this 
plan  was  found  to  be  open  to  two  objections  :  the  lightest 
wind  agitated  the  baiK1.,  and  the  pulley-guides  tore  it  at  the 
points  of  riveting,  whilst  the  guides  themselves  were 
rapidly  worn  out.  Notwithstanding  these  objections  this 
plan  rendered  valuable  service,  and  continued  in  operation 
for  a  year  and  a  half,  transmitting  12  horse-power  to  one 
hundred  looms. 

"The  difficulties  of  the  flat  band  suggested  round  wire 
ropes  ^  inch  diameter  ;  these  were  accordingly  substituted 
and  were  placed  upon  the  same  wooden  pulleys,  which, 
however,  were  first  grooved  to  the  depth  of  half  an  inch.f 

This  plan  answered  every  expectation,  and  experience 
having  fully  sanctioned  its  use  a  second  wire  rope  was  soon 
put  in  operation,  transmitting  the  power  to  a  distance  of 

*  Mr.  H.  M.  Morrison,  Proc.  Inst.  M.  E.  1874,  p.  57. 

f  Prof.  Unwin,  in  his  Howard  lectures  (see  Electrician,  Feb.  3, 
1893),  states  that  an  English  engineer,  Mr.  Tregoning,  suggested  the 
substitution  of  the  wire  rope. 


OF  THE 


64  ROPE-DRIVING. 

about-  770  feet.  The  two  pulleys  were  each  9  feet  6  inches 
diameter,  making  91|  revolutions  per  minute,  and  a  steel 
rope  |  inch  diameter  was  employed  transmitting  50  h.  p. 
at  a  speed  of  31  miles  per  hour.  In  this  instance  it  was 
found  necessary  to  have  supporting  pulleys  to  prevent  the 
rope  from  trailing  upon  the  ground.  These  carrying  pul- 
leys were  placed  half-way  between  the  transmitting  pulleys, 
or  128  yards  apart,  and  in  the  first  instance  they  occasioned 
very  great  difficulties  by  the  rapidity  with  which  they  were 
worn  out  in  the  groove.  They  were  constructed  succes- 
sively of  copper,  wood,  and  polished  cast  iron,  and  were  also 
faced  with  leather,  horn,  india-rubber,  lignum-vitae,  and 
boxwood.  All  these  failed,  however;  the  facings  were  soon 
worn  out,  and  when  the  groove  was  of  metal  or  hard  wood 
and  did  not  itself  wear  it  destroyed  the  rope.  After  re- 
peated experiments  a  dovetailed  groove  was  formed  in  the 
bottom  of  the  pulley  groove  and  filled  with  gutta-percha, 
(as  shown  in  Fig.  65,  page  186.) 

"  This  turned  out  a  perfect  success,  and  carrying  pulleys 
thus  faced  have  an  almost  unlimited  amount  of  durability/' 
Fibrous  ropes  were  used  in  the  United  States  for  long  dis- 
tance transmissions  a  few  years  later  ;  thus  we  find  a  com- 
munication in  the  Scientific  American*  in  which  a  cor- 
respondent from  Winsted,  Conn.,  speaks  of  several  rope- 
drives  in  his  vicinity,  one  of  which  had  been  in  use  since 
1858.  "It  transmits  the  power  for  a  manufactory,  em- 
ploying several  circular  saws,  across  the  river,  225  feet  dis- 
tant, by  a  |-inch  rope  running  over  two  pulleys  six  feet  in 
diameter,  at  a  speed  of  5600  per  minute.  The  pulleys  are 
sheltered,  but  the  rope  runs  exposed  in  all  kinds  of 
weather,  needing  no  attention  except  at  times  to  be  rubbed 
with  grease  having  a  very  small  amount  of  rosin  mixed 
with  it." 

*  Vol.  HI,  1861,  p.  315, 


ROPE-DRIVING. 


65 


<   » 


66  ROPE-DRIVING. 

When  power  is  taken  from  a  water-wheel  in  locations 
where  laud  is  not  available  for  buildings,  the  use  of 
ropes  as  a  means  of  transmitting  the  power  from  the 
wheel  to  the  mill  or  factory  forms  a  most  economical  ar- 
rangement if  the  drive  is  properly  designed  for  the  work. 
It  is  a  great  advantage  in  many  other  cases  to  have  the 
power  plant  and  the  several  buildings  of  a  works  isolated 
from  each  other;  this  is  especially  desirable  in  sawmills 
and  wood -working  establishments,  where  the  risk  from  fire 
is  greatly  reduced  by  such  an  arrangement. 

An  example  of  rope-driving  in  which  the  conditions  are 
particularly  adapted  to  this  form  of  transmission  was 
erected  a  few  years  ago  at  Portland,  Ore.  The  mill  is  built 
on  piles  situated  in  the  Willamette  River,  while  the  engine 
and  boiler  room  are  upon  the  solid  ground,  some  distance 
away.  The  engines  are  a  pair  of  Wheelocks,  with  cylinders 
32  by  60  inches,  intended  to  be  speeded  to  70  revolutions 
per  minute.  The  fly-wheel  is  24  feet  in  diameter  by  66 
inches  face;  it  is  built  up  of  ten  segments  grooved  for 
thirty-three  1^-inch  manilla  ropes,  and  fitted  to  a  shaft  20 
inches  diameter;  its  weight  is  40  tons. 

Two  ropes  are  taken  from  this  wheel  to  jack-shafts  35 
and  45  feet  distant;  the  driven  pulleys — one  with  16  and 
the  other  with  17  wraps — are  each  76  inches  in  diameter, 
and  are  keyed  to  10-inch  shafts.  From  the  end  of  each 
jack-shaft  a  600-h.-p.  transmission  is  arranged  and 
carried  to  the  mill.  Each  shaft  is  fitted  with  a  friction 
clutch  to  allow  either  of  the  mill  transmissions  to  be  thrown 
out  if  desired. 

The  general  arrangement,  showing  location  of  driving 
and  driven  sheaves,  is  represented  in  Figs.  34  and  35,  from 
which  it  is  seen  that  power  is  delivered  to  two  shafts  7 
inches  in  diameter — in  the  one  case  at  a  diagonal  distance 
of  about  200  feet,  and  in  the  other  at  a  distance  of  185  feet 
from  their  respective  drivers — both  the  driven  shafts  being 


ROPE-DRIVING.  67 

at  right  angles  with  the  jack-shaft.  Each  drive  was 
designed  to  transmit  600  h.  p.  with  a  rope  velocity  of  7550 
feet  per  minute,  but  it  has  since  been  found  advantageous 
to  reduce  this  speed  to  about  6000  feet  per  minute. 

The  arrangement  of  ropes  in  these  transmissions  is 
similar  to  that  used  in  the  Chicago  City  Railway  Company, 
illustrated  on  page  53.  In  the  present  case  only  three  wraps 
of  1^-inch  rope  are  used  to  convey  the  power  from  the  jack- 
shaft  to  the  mill,  but  in  order  to  prevent  slip  and  decrease 
the  tension  in  the  slack  part  of  the  rope  the  driving  and 
driven  pulleys  have  each  nine  grooves,  six  turns  being 
carried  around  another  pulley  or  winder,  thereby  increas- 
ing the  arc  of  contact  and,  hence,  the  adhesion.  In  this 
plant  the  stress  in  the  rope  is  very  much  greater  than  that 
ordinarily  used. 

It  is  evident  that  the  whole  strain  must  be  borne  by  the 
three  strands,  as  it  is  only  the  difference  in  tension  of  the 
tight  and  slack  sides  of  the  ropes  that  can  be  used  to  trans- 

PV 

mit  power;   since  =  h.  p.,  we  find  the  difference  in 

ooOOO 

33000  X  600 

tension,  Jr  = ^7Tn —       ~  3300  pounds  ;  and  as  there 

oOOO 

are  three  wraps,  the  difference  in  the  stresses  in  the  two  por- 
tions of  the  rope  will  be  1100  pounds.  As  we  shall  find 
later  the  total  stress  will  be  greater  than  this,  due  both  to 
the  action  of  centrifugal  force  in  the  rope  and  to  the  force 
necessary  for  adhesion. 

As  the  maximum  working  tension  for  a  1^-inch  rope  is 
usually  only  about  450  pounds,  it  is  evident  that  each 
wrap  carries  nearly  three  times  its  proper  load,  taking  the 
wear  and  life  of  the  rope  into  account.  The  engine  has 
only  developed  about  700  horse-po-  er  as  yet,  so  the  total 
stress  in  each  rope  has  been  very  much  less  than  the 
above, — probably  not  more  than  300  to  350  horse-power  on 


68  HOPE-DRIVING. 

each  drive;  but  even  with  this  reduced  stress  one  rope  was 
replaced  inside  of  fifteen  months. 

For  short  distances  shafting  is  often  employed  for  such 
transmissions,  but  with  this  latter  the  friction  of  the  jour- 
nal-bearings is  a  very  important  consideration,  and  effectu- 
ally debars  its  use  for  long  distance  transmission. 

This  will  be  seen  from  the  following  considerations : 

Let  0  =  distortion  of  shaft  (circular  measure)  per  unit 

length; 

0°  =  distortion  in  degrees; 
I  =  unit  length  of  shaft; 
L  =  length  of  shaft  in  feet; 

r  =  distance  of  outer  fibres  from  axis  =  -; 

Z 

d  —  diameter  of  shaft; 
PR  =  twisting  moment  on  the  shaft; 
JV"  =  revolutions  of  shaft  per  minute; 
v  =  velocity  of  circumference  of  shaft  =  ndN\ 
G  =  modulus  of  torsion  of  the  material 

=  two  fifths  of  the  modulus  of  elasticity; 
f  =  maximum  torsional  stress  in  the  outer  fibres 
IQPIt 

TTd3      ' 

W  =  weight  of  shaft  —  3.36  pounds  per  foot  per  square 

inch  of  section ; 
F  —  load  due  to  friction. 

ft       WPRl  m 

Then  *  =        =__ (i) 


If    the    angle    of    torsion    is    given    in  degrees,  then 

_  *°  X  2*       , 
~360~ 

of  length  will  be 


0°    X   %7C 

6  = ;  therefore  the  angular  distortion  per  foot 

360 


ROPE-DRIVING.  69 


0o  _  360   X/£X12_360/      1ZL 

'  Tn    X   ~Gr~    -~^G  '    ~T-    '    '     (2> 

The  working  limit  of  the  angle  of  torsion  for  steel  shaft- 
ing ought  not  to  exceed  0  10  degree  per  foot  in  length  of 
the  shaft;  that  is,  6°  =  0.10Z,  and  for  wronght-iron  shaft- 
ing fl°  =  0.075  L-*  assuming  the  shaft  to  be  of  steel  and 
substituting  the  corresponding  value  for  8°  in  (2),  we  obtain 

0  WL  X  nGd  =  360/  X  12Z; 
hence  /-  800<Z  if  we  assume  that  G  —  11,000,000  pounds. 

Since  the  horse-power  transmitted  by  the  shaft  equals 

PR  x  ~     rpr,  if  we  substitute  the  value  of  PR,  (=  -r^-/), 
ooOOO  \       16'  I 


Ttd*      2 

there  is  obtained  h.p.  =  -  --  f  ;  but  the  velocity  at 

16  '    ooOOO 

the  circumference  of  the  shaft  is  v  =  ndN,  also/=  800^; 
hence 

h.p.  =  0.0095<Fv  .......  (3) 

If  the  bearing  is  well  worn  and  fitted  to  its  shaft  the 
resistance  due  to  friction  will  probably  lie  between  the 

limits  -(pW  and  ~<f>W,\    or   between  1.570  Tf  and    1.28 

&  Tt 

01XF,  where  0  is  a  coefficient,  which  in  the  present  case  we 

shall  assume  equal  to  0.06. 

4. 
Taking    the    lesser  value,    we   shall   have   F=  ~0W, 

where  .Pis  the  force  at  the  circumference  of  shaft  i  eces- 
sary  to  overcome  the  journal   friction.     If  there  are  no 

pulleys  on  the  shaft  W=  ~^L  X  3.36;    the   horse-power 

exerted  to  overcome  the  friction  will  then  be 

Fv          4  ,       TT  d*L  X  3.36?' 
h'P"  =  33000  =  n*  X  4  ~- 


*  Reuleaux  :  Der  Konstrukteur. 
f  Unwin. 


0 


ROPE-DRIVING. 


pressed  as  a  ratio,  the  percentage  of  power  required  to 
overcome  friction  will  be 


&•   •     •    (4) 


Mb  =  o. 

h.p.        0.0095dV 
from  which  there  is  obtained 

h.p.0  =  0.00063^  X  h.p.  - 
d 

That  is,  for  a  steel  shaft  whose  diameter  is  one  inch  the 
horse-power  required  to  overcome  the  friction  in  a  length 
of  1585  feet  will  be  equal  to  the  total  allowable  transmit- 
ting capacity  of  the  shaft. 

For  wrought  iron  6"  ~  .075 L  and  h.p.  —  .0075«T0,  from 
which  may  be  determined  the  value  of  the  ratio 


=  0.00( 


or    hp..=fx^0..     -(5) 


The  following  tables,  calculated  from  these  formulae, 
give  the  limits  at  which  the  power  transmitted  by  a  shaft 
is  absorbed  by  the  friction  of  the  bearings,  the  assumption 
being  that  the  factor  of  journal-friction  0  equals  0.06,  and 
that  the  allowable  twist  shall  not  exceed  0.10  degree  per 
foot  of  length  for  steel,  nor  .075  degree  for  wrought  iron. 
The  shaft  is  supposed  to  be  free  from  all  pulleys  and  gears. 

TABLE  I.  —  LIMIT  OF  LENGTH  FOR  STEEL  SHAFTING. 
No  pulleys  on  the  line. 


Diameter  of 
shaft  in 
inches. 

Length  in  feet 
when  total 
power  is  absorbed. 

Length  when 
efficiency  = 
50  per  cent. 

Length  when 
efficiency  = 
75  per  cent. 

1 
2 
3 
4 
5 

1,585 
3,170 
4,755 
6,340 
7,925 

792 
1,585 
2,377 
3,170 
3,962 

396 
792 

1,188 
1,585 
1,981 

ROPE-DRIVING. 


71 


TABLE  II. — LIMIT  OF  LENGTH  FOR  WROUGHT-IRON  SHAFTING. 

No  pulleys  on  the  line. 


Diameter  of 
shaft  in 
inches. 

Length  in  feet 
when  total 
power  is  absorbed. 

Length  when 
efficiency  — 
50  per  cent. 

Length  when 
efficiency  = 
75  per  cent. 

1 
2 
3 
4 
5 

1,250 
2,500 
3,750 
5,000 
6,250 

625 
1,250 

1,875 
2,500 
3,125 

312 
625 
937 
1,250 
1,562 

From  the  foregoing  it  will  be  seen  that  shafting  is  alto- 
gether unsuitable  for  conveying  power  any  considerable 
distance,  and  as  belting  is  not  adapted  to  this  work  choice 
must  be  made  of  some  other  method. 

For  a  mere  dead  pull,  such  as  the  alternate  strokes 
needed  to  operate  a  pump,  work  is,  and  has  long  been, 
transmitted  to  very  great  distances;  as  by  the  long  lines  of 
draw-rods,  ropes,  or  wires  used  in  mining  regions,  quarries, 
and  elsewhere,  for  transmitting  the  power  of  a  water-wheel 
by  means  of  a  crank  on  its  main  axis,  pulling  during  half 
its  revolution,  against  a  heavy  weight  at  the  end  of  the 
line,  and  thus  storing  up  energy  for  the  return  stroke. 

Wooden  pump-rods  were  used  in  this  manner  about  1865 
near  Petroleum,  W.  Va.  A  large  condensing  engine  was 
located  in  a  central  position,  and  the  rods  transmitted  the 
power  to  a  number  of  oil-wells,  twenty-seven  in  all,  situated 
at  various  distances  and  in  different  directions  from  the 
source  of  power.  The  greatest  distance  was  about  four 
miles. 

Posts  with  crank-arms  were  used  to  change  the  direction 
of  the  pull.  The  rods  were  of  hickory,  connected  end  to 
end  by  means  of  iron  straps. 

The  transmission  of  power  from  the  famous  72-foot  di- 
ameter overshot  wheel  at  Laxey,  on  the  Isle  of  Man,  is  by 
means  of  similarly  connected  trussed  rods,  which  in  this 


72  ROPE-DRIVING. 

case  are  supported  at  regular  intervals  on  small  wheels 
running  on  iron  ways.  About  150  horse-power  are  trans- 
mitted in  this  way. 

This  method  was  adopted  on  a  very  large  scale  in  the 
mines  of  Devonshire  for  the  transmission  of  power  from 
large  overshot  water-wheels  to  pumps  fixed  in  the  shaft  of 
the  mine  at  a  considerable  distance  higher  up  the  valley. 

In  one  case*  the  water-wheel  was  52  feet  diameter,  12 
feet  breast,  and  its  ordinary  working  speed  was  5  revolu- 
tions per  minute.  The  length  of  stroke  given  by  the  crank 
to  the  horizontal  or  "flat "  rods  was  8  feet;  the  rods  were 
3|-inch  round  iron,  and  were  carried  on  cast-iron  pulleys. 

At  Devon  Great  Consols,  near  Tavistock,  there  are  alto- 


£lec.  World 

Fra.  36.— ROPE-DRIVES  WITH  BENT  CRANKS  AT  120  DEGREES. 

gether  very  nearly  three  miles  of  3-inch  wrought-iron  rods, 
carried  on  bobs,  pulleys,  and  stands,  whereby  power  for 
pumping  and  winding  is  conveyed  along  the  surface  to  dif- 
ferent parts  of  these  extensive  mines,  from  11  large  water- 
wheels  ranging  up  to  50  feet  in  diameter,  to  which  the 
water  is  brought  along  eight  miles  of  leats  18  feet  in  width. 

Rods  and  wire  ropes  have  also  been  used  to  transmit  ro- 
tary motion  to  a  considerable  distance  in  a  similar  manner 
by  placing  the  cranks  at  120  degrees,  as  shown  in  Fig.  36. 

It  is  evident  that  the  distance  of  transmission  by  this 
contrivance  will  be  subject  to  the  sag  of  the  ropes,  unless 

*  "  The  Old  Wheal  Friendship  Mine,  near  Marytavy."  Proc.  lust. 
M.  E.  1881,  p.  100. 


OP  THE 


ROPE  DRIVIK>!>*       73 


intermediate  shafts  are  employed.  The  motion  must  also 
be  comparatively  slow,  owing  to  the  severe  strains  which 
would  be  thrown  upon  the  bearings  and  pins  by  the  surg- 
ing and  swaying  of  the  ropes  during  the  rapid  changes  of 
motion  to  which  they  would  be  liable.  In  order,  then,  to 
transmit  much  power,  heavy  rods  or  large  ropes  would  be 
necessary,  and  under  these  conditions  economical  transmis- 
sion would  be  limited  to  short  distances. 

Among  the  various  means  in  use  at  the  present  time  for 
conveying  power  to  a  distance  we  find  steam,  water,  gas, 
compressed  air,  electricity,  and  rope  systems.  Each  of 
these  has  its  own  applications  and  advantages,  but  it  must 
be  borne  in  mind  that  with  the  exception  of  rope  trans- 
mission, of  which  numerous  examples  have  already  been 
given,  all  other  forms  usually  require  a  generator  at  the 
one  end  and  a  motor,  with  separate  attendants,  at  the  other. 

Other  things  being  equal,  the  relative  merit  of  various 
methods  of  transmitting  power  will  be  indicated  by  the 
cost  of  transmitting  a  certain  amount  of  power  to  any 
given  point,  as  compared  with  the  cost  of  this  power  at 
the  generating  station,  while  their  absolute  merit  will  be 
shown  by  comparing  the  cost  of  the  transmitted  power  at 
the  receiving  station  with  the  cost  of  producing  the  re- 
quired power  directly  at  this  point.*  Such  determinations 
are  materially  affected  by  variations  in  the  amounts  of 
power  and  in  the  distance  of  transmission;  the  other  prin- 
cipal factors  to  be  considered  being  the  efficiency  of  the 
system,  the  number  of  working  hours  per  annum,  the 
price  of  1  h.  p.  per  hour  at  the  generating  and  receiving 
stations,  and  the  convenience  and  applicability  of  the  sys- 
tem to  each  special  case. 

The  efficiency  of  any  system  of  transmitting  power  is 
expressed  by  the  ratio  of  the  power  obtained  at  the  receiv- 


*Stahl,  "Wire-rope  Transmission." 


74  ROPE-DRIVING. 

ing  station  to  the  power  given  out  at  the  generating  sta- 
tion. In  all  systems  losses  of  power  of  greater  or  less 
magnitude  occur,  and  the  most  efficient  system  is  that  in 
which  those  losses  are  reduced  to  a  minimum.  We  shall 
not  attempt  here  to  lay  down  rules  governing  the  choice  oi' 
any  particular  method,  for  the  requirements  and  condi- 
tions are  so  varied  that  every  individual  case  must  be 
decided  upon  by  the  engineer  separately  with  a  knowledge 
of  all  the  facts  before  him.  Our  present  object  is  to  ascer- 
tain the  principles  governing  the  use  of  ropes,  and  to 
determine  those  conditions  best  suited  to  their  economic 
working. 


ROPE-DRIVING.  75 


CHAPTER   V. 

THE  subject  of  rope-driving  may  properly  be  placed 
under  two  heads,  according  to  the  nature  of  the  material 
composing  the  ropes,  whether  metallic  or  non-metallic. 
With  few  exceptions  metallic  or  wire  ropes  are  used  almost 
exclusively  on  long-distance  or  telo-dynamic  transmission, 
while  non-metallic  ropes  are  employed  for  intermediate  and 
comparatively  short  drives,  the  consideration  of  which  con- 
stitutes the  present  subject-matter.* 

Among  the  materials  employed  in  this  method  of  power 
transmission  we  find  special  forms  of  leather  belting  used 
as  ropes  working  in  V  grooves;  fibrous  ropes,  including  flax, 
hrmp,  cotton,  and  manilla,  are,  however,  chiefly  employed. 

Rawhide  ropes,  which  are  made  from  f  inch  to  2  inches 
in  diameter,  are  used  to  a  limited  extent.  Where  the 
stress  in  the  rope  is  not  great  and  the  accompanying  slip 
is  small,  rawhide  works  very  well,  and  will  last  from  three 
to  six,  and  in  some  cases  ten,  years.  Under  ordinary  cir- 
cumstances it  is  not  necessary  to  use  any  dressing,  as  suf- 
ficient lubrication  is  furnished  by  the  rope  itself;  if  the 
rope  slips  in  its  groove  the  leather  will  be  burned,  and  lose 
its  flexibility,  and  also  its  adhesive  qualities,  to  a  certain 
extent.  A  rawhide  rope  has  very  little  tendency  to  rotate 

*  Concerning  wire-rope  transmission  the  reader  is  referred  to  the 
following: 

"Wire-rope  Transmission"  (A.  W.  Stnhl);  "Elektrischen  Kraft 
iibertragung"  (A.  Beringer);  "Drahtseiltriebs"  (D  H.  Ziegler); 
"  Constructeur"  (F.  Reuleaux);  "Machine  Design"  and  "Central 
Stations"  (W.  C.  Uuwin).  Also  trade  pamphlets  published  by  W. 
A.  Roebliug  &  Sons;  Cooper,  Hewitt  &  Co.;  and  others. 


76  HOPE-DRIVING. 

on  its  axis;  for  this  reason  the  wear  is  not  always  uniform, 
and  with  a  heavy  tension  it  is  liable  to  take  the  set  of  the 
groove  in  which  it  runs.  This  is  rather  an  advantage  for  a 
straight  drive,  where  the  rope  always  runs  in  the  same  di- 
rection; but  in  those  cases  where  a  rope  is  led  on  to  the 
pulleys  at  an  angle  this  will  be  a  disadvantage,  as  under 
such  conditions  the  rope  often  slips,  and  wear  is  excessive. 
Where  the  rope  is  subject  to  wet  or  dampness,  rawhide  is 
an  excellent  material  to  use,  as  it  is  very  little  affected  by 
dampness.  The  cost  of  rawhide  rope  will  average  about 
six  times  that  of  a  good  quality  of  manilla  transmission- 
rope,  and  although  it  is  to  be  preferred  in  certain  cases,  its 
greater  cost  will  limit  its  application. 

Round-leather  ropes,  formed  by  twisting  narrow  strips  of 
leather  into  a  continuous  spiral,  are  used  for  light  driving, 
and  are  very  desirable  for  some  classes  of  work. 

Solid  round-leather  ropes,  made  from  several  thicknesses 
of  belting  cemented  together  and  secured  with  screwed 
wire  forced  into  the  leather,  are  made  in  various  sizes  up 
to  2^  inches  in  diameter,  but  sizes  larger  than  f  or  1  inch 
in  diameter  are  seldom  used. 

Steel  ropes  with  leather  washers  closely  threaded  on 
have  been  tried  with  considerable  success,  but  the  expense 
of  such  a  rope  would  necessarily  limit  its  application. 

Other  special  forms  of  leather  belting  used  as  ropes  are 
found  in  the  various  modifications  of  the  square  and  angu- 
lar belts  which  have  been  used  for  a  number  of  years  for 
both  light  and  heavy  drives. 

Leather  ropes  as  large  as  If  inches  square,  made  up  of 
layers  of  leather  cemented  together  so  that  the  whole  is 
uniform  and  continuous,  have  been  used  to  replace  quar- 
ter-turn flat  belts,  and  also  for  main  driving. 

These  run  in  V  grooves  so  that  the  adhesion  is 
greatly  in  excess  of  that  produced  by  a  flat  belt  on  a 
smooth  pulley  under  the  same  tension.  In  the  same 


ROPE-DRIVING.  77 

way  triangular  belts  built  up  from  various  thicknesses  of 
leather  possess  the  advantages  characteristic  to  all  forms 
of  rope-driving  which  use  a  V  groove,  viz.,  greater  ad- 
hesion for  a  given  tension,  and  the  facilit}-  with  which  such 
transmitters  lend  themselves  to  the  communication  of 
power  between  shafts  at  an  angle  with  each  other. 

It  is  evident  that  several  of  these  leather-rope  belts  may 
be  used  side  by  side  in  a  manner  similar  to  the  various  ap- 
plications of  fibrous  ropes.  Such  ropes  have  proven  satis- 
factory in  those  cases  where  the  pulleys  are  of  approxi- 
mately the  same  diameter;  but  on  pulleys  whose  diame- 
ters vary  considerably  each  portion  of  the  leather  rope  in 
contact  with  the  driver  tends  to  rotate  the  follower  at  a 
different  velocity,  necessarily  producing  slip  and  wear,  to 
an  extent  depending  upon  the  ratio  of  the  diameters  em- 
ployed. 

In  England  manilla  is  now  being  used  very  largely,  but 
cotton  ropes  were  formerly  preferred  to  the  exclusion  of  all 
others  for  all  kinds  of  driving;  but  the  most  probable  cause 
of  this  was  not  that  cotton  was  the  best  or  most  economi- 
cal material  for  the  purpose,  but  that  rope-driving  is  most 
common  at  cotton  factories,  and  cotton  ropes  were  made 
in  the  locality  by  men  who  were  familiar  with  the  local 
product,  and  had  been  for  years  making  spindle  and  rim 
bands  of  small  size.  When  the  demand  for  large  sizes 
arose  these  rope-makers  applied  themselves  to  the  newer 
industry,  and  shut  out  other  materials.* 

In  the  mills  of  Dundee  and  vicinity,  and  in  the  North  of 
Ireland,  where  flax  and  hemp  are  worked,  we  find  ropes  of 
hemp,  a  local  product,  used  entirely. 

In  many  cases  ropes  of  cotton  are  to  be  preferred,  as  they 
are  generally  softer  and  more  pliable  than  the  ordinary 
manilla  ropes,  thus  allowing  smaller  pulleys  to  be  used 

*  W.  H.  Booth,  Am.  Machinist,  January,  1891. 


78  ROPE-DRIVING. 

with  less  injury  to  the  fibres.  In  fact,  cotton  ropes  of 
small  diameter  have  been  used  for  years  in  cotton  ma- 
chinery bandings  over  pulleys,  and  under  conditions  which 
would  wear  out  a  manilla  rope  in  one  third  the  time. 
There  is  also  an  advantage  in  that  there  is  less  internal 
chafing  and  wear  when  the  rope  is  bent  over  a  pulley,  on 
account  of  the  smoothness  of  the  fibres  and  the  great  elas- 
ticity of  the  yarns. 

The  cotton  fibre  is  not,  as  it  appears  to  the  eye,  a  solid 
cylindrical,  gossarnerlike  hair,  but  when  fully  ripe  is  shown 
under  the  microscope  as  a  flattened  hollow  ribbon  or  col- 
lapsed cyliudric  tube  twisted 
several  times  throughout  its 
length,  as  shown  in  Fig.  37;* 
it  is  of  equal  size  for  about 
three  fourths  of  its  length,  and 
it  then  gradually  tapers  to  a 
point.  This  point  is  a  section 
almost  perfectly  cylindric,  and, 
unlike  the  rest  of  the  fibre, 
often  composed  of  solid  matter. 

Covering    the    outside    mem-  _ 

,  f  .  FIG.   37.— COTTON  FIBRE,   OR- 

brane  of  the  fibre  is  an  oleagi-     LEANS   VARIETY  (Oossypium 

nous  coating  generally  known  Hirsutum). 
as  cotton-wax.  This  wax  amounts  to  about  two  per  cent 
of  the  fibre,  and  in  the  spinning  of  the  material  it  requires 
to  be  reduced  to  a  certain  point  of  liquefaction  by  the 
heated  temperature  of  the  room  before  it  can  be  made  to 
work  properly  without  lapping  on  the  drawing  rollers. 

These  fibres  vary  in  size  from  0.00084  inch  mean  di- 
ameter, and  about  \  inch  long  to  0.000635  inch  mean 
diameter  and  2£  inches  in  length,  depending  upon  the 
variety  of  the  cotton;  but  for  a  given  variety  the  differ- 

*  See  "The  Cotton  Fibre,"  by  Hugh  Monie,  Jr.  Published  by 
Hey  wood  &  Sou,  Manchester,  Eng. 


ROPE-DRIVING.  79 

eo»e  is  very  small:  thus  in  the  Sea  Island  cotton  the 
maximum  length  of  fibre  is  2  inches,  while  the  minimum 
is  If  inches;  in  the  same  way  in  the  Orleans  variety 
shown  in  Fig.  37  the  maximum  length  is  1^  inches  and 
the  minimum  -}-|inch. 

As  a  rule,  those  cottons  which  have  the  longest  fibres 
are  also  the  smallest  in  diameter:  they  possess  the  natural 
twist  in  a  more  perfect  and  highly  developed  form,  and 
are  much  stronger  and  more  elastic. 

In  all  good  commercial  fibres  of  cotton  there  is  neces- 
sarily (1)  a  very  small  percentage  of  solidified  oleaginous 
matter  distributed  over  the  internal  surface  of  the  fibre 
deposited  when  the  vital  fluids  were  in  active  circulation; 
and  (2)  a  certain  percentage  of  moisture  known  as  water 
of  hydration. 

These  together  with  the  twisted  structure  impart  to  the 
fibres  that  suppleness,  tenacity,  and  elasticity  without  which 
they  would  be  almost  useless  for  manufacturing  purposes. 

The  cotton  fibre  is  thus  naturally  well  adapted  to  the 
work  of  being  twisted  into  yarns;  the  presence  of  the 
natural  convolutions  and  comparative  smoothness  of  the 
surface  of  the  unit  filament  permits  considerable  elonga- 
tion, and  the  wax  on  its  surface  serves  as  a  natural  lubri- 
cant and  prevents  the  fibres  from  becoming  brittle. 

Thus  it  will  be  seen  that  ropes  made  from  fibres  pos- 
sessing these  characteristics  are  particularly  well  adapted 
to  the  transmission  of  power  in  which  the  rope  is  con- 
stantly undergoing  a  varying  strain  and  is  subjected  to 
much  flexion. 

The  strength  of  cotton  ropes  is,  however,  relatively  small 
when  compared  with  other  fibrous  ropes,  and  although  the 
weight  is  about  one  fifth  less  than  manilia,  for  equal  di- 
ameters, the  actual  first  cost  is  from  fifty  to  seventy  five 
per  cent  greater  than  for  the  latter. 

Nystrom  gives   the   breaking  strength  of   three-strand 


80  ROPE-DRIVING. 

cotton  ropes  at  less  than  one  tenth  that  of  similar  manilla 
rope,  but  this  is  apparently  too  low  for  a  good  quality  of 
transmission  rope. 

Tests  made  at  Watertown  on  a  number  of  Lambeth 
ropes  varying  in  size  from  1  inch  to  2£  inches  in  diameter 
indicate  that  the  breaking  strength  is  equal  to  about 
4000tr  pounds,  while  the  extension  varies  from  twenty  to 
twenty-five  per  cent,  corresponding  to  a  reduction  in 
diameter  of  about  fifteen  per  cent.  The  weight  of  these 
ropes  is  very  closely  0.26(f  pounds  per  foot  of  length. 

A  series  of  tests  carried  out  by  Kircaldy  *  on  cotton 
ropes  ranging  in  size  from  ^-f  inch  to  2T\-  inches  in  di- 
ameter give  the  breaking  strength  as  3700d2  pounds  for 
minimum  value  and  5800eP  pounds  as  a  maximum.  The 
weight  per  foot  of  these  ropes  varied  from  0.25d'2  to  0.29cP  ; 
the  extension  under  a  stress  of  about  85  per  cent  of  the 
breaking  load  varied  from  17  to  27  per  cent. 

Reduced  to  a  common  basis  in  which  the  strength  is 
made  proportional  to  the  weight,  and  averaging  the  results, 
we  find  that  the  breaking  strength  may  be  represented  by 
4600cT  pounds. 

The  data  on  cotton  ropes  are  too  meagre  to  determine 
whether  their  strength  decreases  as  the  diameter  increases, 
but  this  is  probably  the  case. 

Reuleaux  gives  7500  pounds  per  square  inch  of  section 
for  cotton  transmission-ropes,  which  agrees  very  closely 
with  the  above  values. 

From  the  formula,  breaking  strength,  S  =  4600^3  pounds 
the  values  given  in  Table  III  have  been  calculated,  and 
may  be  considered  as  representing  approximately  the 
strength  of  cotton  transmission-ropes  of  good  quality. 

The  working  strength  of  cotton  transmission-rope  may 
be  taken  higher,  in  proportion  to  its  ultimate  strength, 

*See  also  Kent's  "  Mechanical  Engineer's  Pocket  Book." 


HOPE-DRIVING.  81 

TABLE  III. — STRENGTH  OF  COTTON  TKANSMISSION-ROPES. 

Ultimate  Brea^in^Strength, 

*  1,150 

|  1,800 

f  2,600 

%  3,500 

1  4,600 
li  7,200 
11  10,400 
If  14,000 

2  18,400 

than  is  used  with  manilla,  for  the  latter  is  weakened  by 
the  grease  with  which  it  is  lubricated;  and,  moreover,  a 
larger  factor  must  be  allowed  for  wear  on  account  of  the 
character  of  the  manilla  fibre,  which  breaks  more  easily 
under  bending  strains. 

As  compared  with  manilla,  then,  the  advantages  of  cot- 
ton ropes  of  the  same  diameter  are:  Greater  flexibility, 
greater  elasticity,  less  internal  wear  and  loss  of  power  due 
to  bending  the  fibres,  and  the  use  of  smaller  pulleys  for  a 
given  diameter  of  rope.  Its  disadvantages  are:  Greater 
first  cost,  lesser  strength,  and,  possibly,  a  greater  loss  of 
power  due  to  pulling  the  ungreased  rope  out  of  the  groove 
—in  any  case  this  is  usually  small  with  speeds  over  2000 
feet  per  minute, 

As  we  have  already  noted,  manilla  rope  is  used  very  ex- 
tensively for  transmission  purposes,  but  its  application  has 
not  always  met  with  that  success  which  would  follow  u 
more  thorough  knowledge  of  its  requirements.  Inefficient 
rope-drives  are  erected  and  run  for  a  few  months,  or  per- 
haps only  days,  and  are  replaced  with  larger  ropes  if  the 
sheaves  will  permit,  or,  as  in  many  cases,  tne  ropes  give 
way  to  leather  belting,  and  henceforth  rope-driving  is  con- 
demned. The  true  cause  is  not  so  much  the  inefficiency 
of  the  ropes  as  it  is  the  lack  of  knowledge  governing  their 


ROPE-DRIVIKG. 


use  and  application;  in  order  to  obtain  a  proper  concep- 
tion of  this  a  study  of  the  structure  of  the  rope  will  be 
found  advantageous.  Manilla,  or,  more  prop- 
erly, manilla  hemp  (abaca)  rope,  is  made  from 
the  fibres  of  the  Musa  text  His,  a  plant  closely 
allied  to  the  banana,  growing  near  Manilla, 
in  the  Philippine  Islands.  The  fibres  are  a 
part  of  the  outer  covering  of  the  leaf-stalk, 
which  attains  a  length  sometimes  as  great  as 
15  feet.  To  obtain  fibres  of  suitable  size  for 
manufacturing  rope  the  leaf  stalks  are  sub- 
divided, and  in  the  process  of  segregation 
the  fibre  assumes  an  appearance  somewhat 
similar  to  that  produced  in  splitting  a  piece 
of  wood — it  is  rough  and  uneven,  and  more 
or  less  splintery  throughout  its  length.  These 
fibres,  although  in  themselves  not  very  large, 
are  composed  of  very  fine  and  much  elongated  \ 
bast-cells,  which  overlap  each  other  as  shown 
in  Fig.  38.  The  cells  are  irregular  in  outline 
and  vary  considerably  in  size.  The  length 
of  the  cells  is  about  one  fourth  of  an  inch. 
A  series  of  tests  on  mauilla  fibre  carried  out 
by  Dr.  Stanley  M.  Coulter  of  Purdue  Univer- 
sity, shows  that  the  cells  are  not,  as  com- 
monly supposed,  held  together  by  an  inter- 
cellular tissue  or  mucilaginous  substance.  A 
cross-section  of  a  portion  of  the  fibre,  (a), 
Fig.  39,  enlarged  450  times,  shows  that  there 
are  no  intercellular  spaces;  and  various  reac- 
tions to  determine  the  presence  of  mucilage  CELLS 
or  other  vegetable  glue  revealed  no  traces  of  NILLA  FIBRE, 

ENLARGED. 

its  presence. 

The  characteristic  roughness  possessed  by  the  manilla 
fibre  is  due  entirely  to  mechanical  causes,  such  as,  for 


FIG.  38. 


ROPE-DRIVING. 


83 


instance,  the  laceration  of  a  cell  in  the  separation  from  the 
leaf-stalk,  or  the  subsequent  opening  out  of  the  ends  of 
the  cells. 

The  contour  of  the  perimeter  is  also  rough,  as  noted  in 
the  figure,  396,  as  it  retains  the  form  impressed  upon  it  by 
the  contiguous  cells  when  in  the  plant. 

These  fibres  have  great  strength  in  the  direction  of  their 
length,  but  are  weak  transversely;*  when  made  into  rope 
they  are  compelled,  in  bending  over  the  sheave,  to  slide  on 


FIG.     39a.  —  MANILLA    FIBRE, 

MAGNIFIED    450   TlMES. 


0.016* 


FIG.  396.  —  CROSS-SECTION 
OF  MANILLA  FIBRE. 


each  other  while  under  pressure  from  the  load.  This 
causes  the  internal  chafing  and  grinding  which,  if  not  pre- 
vented, soon  wears  out  a  rope  when  subjected  to  bending 
strain. 

In  addition  to  the  action  of  the  fibres  upon  each  other, 
the  strands  and  the  yarns  of  which  the  strands  are  com- 
posed also  slide  a  small  distance  upon  each  other,  causing 
friction,  and  hence  internal  wear. 

By  opening  out  an  old  dry  rope  which  has  been  used  over 
a  sheave,  a  fine  powder  will  be  disclosed,  showing  that 
where  the  rope  was  bent  over  the  sheaves  the  strands,  in 
sliding  on  each  other,  ground  some  of  the  fibres  to  powder. 
Another  reason  for  this  is  that  the  fibres  in  an  old  rope  be- 
come brittle  and  weak  when  dry,  so  that  the  constant 


*  The  tensile  strength  of  manillu  fibres  will  average  over  30,000 
pounds  per  square  inch  of  section. 


84  ROPE-DRIVING. 

flexure  to  which  they  are  subjected  rapidly  disintegrates 
the  cell  congeries. 

Aside  from  the  external  wear  which  a  rope  suffers  from 
contact  with  its  sheave  (part  of  which  is  the  differential 
driving  effect),  these  two  are  the  principal  causes  of  rapid 
wear  in  a  rope-drive,  to  remedy  which  we  must  in  the  first 
case  lubricate  the  fibres,  and  in  the  second,  prevent  undue 
flexure  of  the  rope.  How  this  is  effected  in  practice  will  be 
seen  presently. 


BOPE-DRIVIOT.  85 


CHAPTER   VI. 

Itf  manufacturing  manilla  rope  the  fibres  are  first  spun 
into  a  yarn,,  this  yarn  being  twisted  in  a  direction  called 
right-hand.  From  20  to  80  of  these  yarns,*  depending 
on  the  sizo  of  the  rope,  are  then  put  together  and  twisted 
in  the  opposite  direction,  or  left-hand,  into  a  strand. 
Three  of  these  strands  for  a  3-strand,  or  four  for  a  4-strand, 
rope  are  then  twisted  together,  the  twist  being  again  in 
the  right-hand  direction.  It  will  be  noticed  that  when  the 
strand  is  twisted  it  untwists  each  of  the  threads,  and  when 
the  three  strands  are  twisted  together  into  rope  it  untwists 
the  strands,  but  again  twists  up  the  threads.  It  is  this 
opposite  twist  that  keeps  the  rope  in  its  proper  form. 
When  a  weight  is  hung  on  the  end  of  a  rope  the  tendency 
is  for  the  rope  to  untwist  and  become  longer.  In  untwist- 
ing the  rope  it  will  twist  the  threads  up,  and  the  weight 
will  revolve  until  the  strain  of  the  untwisting  strands  just 
equals  the  strain  of  the  threads  being  twisted  tighter.  In 
making  a  rope  it  is  impossible  to  make  these  strains  exactly 
balance  each  other,  and  it  is  this  fact  that  makes  it  neces- 
sary to  take  out  the  turns  in  a  new  rope — that  is,  untwist 
,  it  when  it  is  put  at  work. 

*  A  three-strand  rope  one  inch  in  diameter  is  the  key  to  the  sizing 
of  the  yarns.  Yarns  of  20s  are  of  such  a  size  as  to  require  20  to  fill 
a  tube  half  an  inch  in  diameter,  or  to  make  one  strand  of  an  inch 
rope;  26s  requires  26  to  fill  the  same  size  tube  ;  and  so  on.  The  size 
of  cotton  yarns,  on  the  other  hand,  depends  upon  the  number  of 
hanks  per  pound  ;  thus  20's,  or  number  29  cotton,  requires  20  hanks 
(340  yards  each)  of  this  size  to  weigh  one  pound. 


86  ROPE-DRIVING. 

The  fibres  of  manilla  which  are  thus  twisted  into  ropes 
will  average  over  6  feet  in  length,  varying  from  3-J-  to  12 
feet.  If  they  were  long  enough,  the  most  advantageous 
method  of  using  them  would  be  to  lay  the  fibres  side  by 
side,  and  secure  them  at  the  two  ends;  each  fibre  would 
then  bear  its  own  share  of  the  strain,  and  the  strength  of 
the  bundle  would  be  that  of  the  sum  of  the  strengths 
of  the  separate  fibres.  As  a  long  rope  could  not  be 
formed  in  this  way,  the  fibres  are  secured  by  twisting 
so  as  to  produce  sufficient  compression  to  prevent  them 
from  moving  upon  each  other  when  a  strain  is  applied ;  but 
in  attaining  this  amount  of  compression  their  strength  is 
greatly  reduced;  this  very  compression  acts  as  a  constant 
weight  on  the  fibre,  and  must  be  deducted  therefrom  before 
the  available  strength  can  be  applied. 

The  weakening  effect  produced  by  twisting  varies  consid- 
erably among  the  fibres  of  the  same  rope  according  to  their 
distance  from  the  centre  or  heart  of  the  bundle.  If  a  cer- 
tain amount  of  twist  be  given  to  a  bundle  of  fibres,  the 
outer  ones  will  be  strained  more,  and  will  act  with  less  use- 
ful effect  than  those  on  the  inside,  which  will  have  to  bear 
the  greater  part  of  the  strain  while  the  rope  is  being  used. 
It  will  therefore  be  evident  that  if  the  fibres  were  twisted 
at  once  into  a  thick  rope  the  outer  fibres  would  be  so  much 
strained  as  to  be  of  little  or  no  use  in  contributing  to  the 
strength  of  the  rope,  but  by  making  the  rope  as  we  have 
indicated  by  first  twisting  into  yarns,  then  into  strands,  and 
finally  combining  these  into  a  rope,  the  strain  is  more 
equalized,  and  the  important  properties  of  length  and 
strength  are  secured  without  too  great  a  sacrifice  of  the 
strength  of  the  individual  fibres. 

The  degree  of  twist  in  the  rope  may  be  determined  by 
constructing  a  right-angled  triangle,  the  base  of  which  is 
the  circumference,  and  the  height  the  length  of  one  turn 
of  the  strand  measured  parallel  to  the  axis.  The  difference 


ROPE-DRIVING.  87 

between  this  height  and  the  hypothenuse  is  the  quantity  by 
which  the  rope  is  twisted.  The  ropemaker's  ordinary  rule 
for  a  three  strand  rope  is  to  have  one  turn  to  as  many 
inches  as  are  contained  in  the  circumference  of  the  rope; 
but  the  degree  of  twist  is  variable  and  more  or  less  de- 
pendent upon  the  judgment  of  the  maker. 

Experiments  by  Reaumur  to  determine  tiie  effect  of  twist 
upon  a  rope  showed  that  a  small  well-made  hemp  cord 
broke  in  different  places  with  a  mean  weight  of  65  pounds; 
while  the  three  strands  of  which  it  was  composed  bore  29^, 
33^,  and  35  pounds  respectively,  so  that  the  total  absolute 
strength  of  the  strands  was  98  pounds,  although  the  aver- 
age real  strength  was  only  65  pounds,  thus  showing  a  loss 
of  33  per  cent. 

More  recently  the  test  of  a  small  rope  showed  an  average 
strength  of  4550  pounds,  while  the  aggregate  strength  of 
its  72  yarns  was  6480  pounds — each  yarn  bearing  about  90 
pounds;  thus  there  is  a  loss  of  1930  pounds,  or  about  30 
per  cent. 

Ropes  made  of  the  same  hemp  and  the  same  weight  per 
foot,  but  twisted  respectively  to  two  thirds,  three  fourths, 
and  four  fifths  of  the  lengths  of  their  component  yarns, 
supported  Uie  following  weights  in  two  experiments  made 
by  Duhamel:* 

Pounds.  Pounds. 

Two  thirds 4,098  4,250 

Three  fourths ...  4,850  6,753 

Four  fifths 6,205  7,397 

The  results  of  these  experiments  led  Duhamel  to  make 
ropes  without  twist  by  placing  the  yarns  together  and 
wrapping  them  round  to  keep  them  together.  The  rope 
had  great  strength  and  pliability,  but  not  much  durability 

*  Tomlinson,  vol.  vn.  p.  574, 

OF  THE 


83  ROPE-DRIVING. 

on  account  of  the  outer  covering  wearing  away  or  opening 
when  bent,  thus  admitting  moisture  to  the  interior,  which 
rotted  the  yarns. 

In  general,  the  greater  the  twist  the  more  hard  and  rigid 
the  rope  is,  and  the  better  it  will  keep  its  form;  but  it  is 
not  as  strong,  weight  for  weight,  as  the  more  loosely 
twisted  rope;  moreover,  the  hard-twisted  rope  is  more  rigid 
than  the  other,  and  is  not  ;is  suitable  for  transmission  pur- 
poses, owing  to  the  rapid  wear  which  constant  flexure 
produces. 

A  very  excellent  transmission-rope,  known  as  the  "  Lam- 
beth," is  made  in  a  somewhat  similar  manner  to  that  just 
described,  but  it  is  not  open  to  the  same  objections.  In 
this  case  the  rope,  which  is  of  cotton,  is  made  of  four 
strands  twisted  together  in  the  usual  manner,  but  the 
strands  are  themselves  composed  of  a  bundle  of  fine  yarns 
and  have  scarcely  an  appreciable  twist.  Each  bundle, 
comprising  many  hundred  yarns,  is  wound  spirally  with 
smaller  bundles  of  about  100  yarns  each.  In  this  way 
the  outside  of  the  rope  acts  as  a  shield  or  covering  to 
the  cores  which  do  the  work.  By  this  means  a  certain 
amount  of  the  natural  elasticity  of  the  cotton  is  re- 
tained, and  its  pliability  is  much  greater  than  in  ordinary 
hawser-laid  ropes. 

Lubrication  of  transmission-ropes  is  provided  for  in 
various  ways.  Frequently  the  rope  is  laid  up  dry  and  a 
coating  or  dressing  is  given  to  the  exterior,  which  is  sup- 
posed to  penetrate  to  the  interior  and  lubricate  the  fibres. 
With  some  dressings  this  may  occur,  but  with  others 
the  effect  is  merely  local;  the  interior  of  the  rope  re- 
mains dry,  and  much  bending  soon  wears  it  out.  With 
cotton  ropes,  as  we  have  already  noted,  the  internal 
chafing  and  wear  is  very  much  reduced,  and  for  this 
reason  cotton  ropes  are  laid  up  dry  and  are  not  usually 


ROPE-DRIVING.  89 

lubricated;  they  are,  however,  generally  coated  with  some 
form  of  dressing  to  prevent  the  fibre  from  rising  or  the 
rope  fraying,  and  to  protect  it  from  an  undue  amount  of 
moisture  when  exposed  to  the  weather;  it  also  assists  in 
retaining  the  natural  moisture  in  the  fibres,  without 
which  they  would  become  brittle  and  weak. 

According  to  the  nature  of  the  dressing,  the  interior 
may  or  may  not  be  affected  by  the  outer  coating.  Bees- 
wax and  black  lead,  with  a  little  tallow,  forms  an  excellent 
moisture-proof  covering  for  ropes;  it  fills  in  the  spaces 
between  the  strands,  and  the  rope  soon  assumes  a  perfectly 
round  and  smooth  appearance,  like  a  bar  of  iron ;  with  this 
coating  the  interior  retains  its  natural  condition,  and  should 
therefore  not  be  used  where  a  lubricant  is  desired.  Pine 
tar  is  much  used  on  cotton  ropes  for  the  same  purpose. 
Mixtures  of  tallow  and  black  lead;  molasses  and  black 
lead;  equal  parts  of  resin  or  beeswax,  with  black  lead, 
tallow,  and  molasses  melted  together,  and  applied  hot;  and 
various  other  compounds  are  in  use  for  this  purpose. 
Tallow,  lard,  and  other  greases  are  used  separately,  and  are 
fairly  satisfactory  as  a  lubricant.  When  the  rope  runs  out- 
of-doors  a  water-proof  coating  is  necessary  to  preserve  it 
from  decay,  and  for  this  purpose,  if  it  is  also  desired  to 
lubricate  the  rope  at  the  same  time,  there  is  probably  noth- 
ing better  than  tallow  and  black  lead,  or  graphite,  pro- 
vided the  rope  is  not  twisted  too  hard,  which  would  pre- 
vent the  dope  from  penetrating.  In  certain  drives  that 
are  subjected  to  hard  service  and  are  more  or  less  exposed 
to  the  weather  boiled  linseed-oil  is  used  very  successfully. 
According  to  Mr.  A.  D.  Pentz,*  the  rope  is  treated  every 
two  weeks  to  about  two  quarts  of  the  oil,  dripped  one  drop 
at  a  time  upon  one  of  the  sheaves,  which  is  uncovered  on 
top;  the  rope  runs  on  the  bottom  of  the  sheave  and  slowly 

*  Eng.  Magazine,  Nov.  1893,  p.  256. 


90  HOPE-DRIVING. 

absorbs  the  oil.  Although  the  rope  is  very  materially 
weakened  by  this  process,  yet  the  greater  freedom  of  the 
fibres  permits  a  heavier  working  strain  to  be  carried,  for  it 
is  the  relative  wear  of  the  fibres  that  determines  the  life  of 
the  rope.  A  manilla  rope  with  the  fibres  properly  lubri- 
cated will,  under  the  same  conditions,  outlast  from  two  to 
four  similar  dry-laid  ropes  which  are  allowed  to  run  dry. 
Manilla  transmission-ropes  are  generally  laid  up  in  tallow? 
paraffine,  soapstone,  or  a  mixture  similar  to  the  above 
preparations. 

A  superior  manilla  transmission-rope  is  that  known  as 
the  "  Stevedore/'  or  black  rope,  which  is  made  with  both 
three  and  four  strands,  the  latter  being  laid  up  about  a 
central  core.  The  yarns  of  this  rope  are  each  coated  with 
a  mixture  of  graphite  and  tallow,  so  that  when  twisted 
into  strands  the  coating  lodges  in  the  hollows  and  uneven 
places  among  the  fibres,  and  thoroughly  lubricates  the 
strands  and  individual  fibres  composing  the  rope,  which  is 
thus  made  practically  as  nearly  water-proof  as  possible. 
After  it  has  been  in  use  a  short  time  its  appearance  is  that 
of  a  black  rod  of  iron,  smooth  and  round,  similar  to  the 
beeswax-coated  rope  previously  mentioned,  but  perfectly 
flexible.  A  manilla  rope  thus  made  will  last  from  three 
to  eight  years  if  not  overstrained;  when  running  indoors 
under  favorable  conditions  the  latter  limit  may  be  attained, 
but  when  exposed  to  the  weather,  or  when  working  under 
less  favorable  conditions,  its  life  will  be  shortened. 

Hemp  ropes  intended  for  outdoor  service  are  sometimes 
treated  by  passing  the  yarns  through  boiling-hot  tar,  suit- 
able machinery  being  used  to  regulate  the  amount  of  tar 
retained  in  the  yarns  so  that  the  fibres  may  be  coated  over 
and  thus  preserved  from  decay.  Tarring  protects  rope 
from  injury  by  exposure  to  rain  and  immersion  in  water, 
but  it  makes  its  fibre  rigid  and  impairs  its  strength;  for 


HOPE-DRIVING.  91 

this  reason  it  is  unsuitable  for  ordinary  transmission  pur- 
poses. 

It  has  been  shown  by  experiment:  * 

.  1.  That  white  or  untarred  rope  in  continual  service  is 
one  third  more  durable  than  tarred.  2.  That  it  retains  its 
strength  much  longer  when  kept  in  stock.  3.  That  it  re- 
sists the  ordinary  injuries  of  the  weather  one  fourth  longer. 

With  the  exception  of  the  outside  yarns  of  large  haw- 
sers, manilla  ropes  are  not  tarred. 

The  breaking  strength  of  a  rope  depends  both  upon 
the  quality  of  the  material  and  the  degree  of  twist  given 
to  the  strands \  for  a  loosely  twisted  rope  of  a  given  diame- 
ter the  strength  is  less  than  that  in  a  hard  twist  of 
the  same  diameter,  but  compared  weight  for  weight,  the 
rope  with  the  lesser  degree  of  twist  is  the  stronger. 

In  discussing  the  strength  of  ropes,  which  formerly  was 
always  given  in  terms  of  the  circumference,  there  is  a  lack 
of  uniformity  among  writers  in  the  relation  between  the 
diameter  and  area  of  a  rope.  The  circumference,  as  meas- 
ured by  a  tape,  depends  upon  the  number  of  strands  in  the 
rope  and  their  compression  upon  one  another.  If  the 
strands  still  retained  their  circular  section  when  twisted 
into  a  rope,  the  circumference  of  a  3-strand  rope  would  be 
2.86  times  the  diameter  of  the  circumscribing  circle,  as 
given  by  Nystrom;  if  the  strands  completely  fitted  the 
circle  its  measure  would  be  n  times  the  diameter  as  given 
by  Unwin  and  others. 

As  neither  of  these  conditions  obtains  in  practice  the 
true  value  lies  between  2.86  and  3.14,  and  we  shall  assume 
3  as  the  most  suitable  factor. 

In  the  same  way  the  area  of  the  cross-section  of  the  rope 
is  variously  estimated  as  that  of  the  area  of  the  circular 
strands,  or  as  the  area  of  the  full  diameter  of  the  rope. 

*Duhamel :  Traite  de  la  fabrique  des  manoeuvres  pour  les  vaisseaux, 


92  ROPE-DRIVING. 

By  assuming  the  rope  to  be  made  of  three  strands,  the 
cross-section  of  each  of  which  is  a  circle,  the  area  of  the 

rope  would  evidently  be  Sf-rO*2)*  where  d  is  the  diameter 

of  each  strand.  If  d  represents  the  diameter  of  rope,  i.e., 
the  diameter  of  circumscribing  circle,  the  area  in  terms  of 
d  will  become 


The  section  of  the  strands,  taken  at  right  angles  to  the 
axis  of  the  rope,  is,  however,  not  a  circle,  as  can  be  seen 
from  Fig.  40.  The  degree  of  twist  given  to  the  strands, 


FIG.  40. — ACTUAL  SECTION  OF  ROPE. 

and  the  compression  of  the  latter  upon  one  another,  will 
evidently  affect  the  area  of  the  section;  for  the  longer  the 
spiral  the  more  nearly  will  the  cross-section  of  each  strand 
approach  a  circle.  It  is  obvious  that  the  true  value  must 
lie  between  0.52<f  and  0.7854eP .  In  determining  this  area 
the  writer  made  a  number  of  plaster  casts  at  different 


ROPE-DRIVING. 


93 


points  of  several  3-strand  manilla  ropes,  varying  in  size 
from  f  inch  to  If  inches  diameter.  With  these  casts  as 
dies,  which  were  covered  with  printer's  ink,  impressions 
were  made  and  the  area  obtained  by  using  a  planimeter. 
The  area  at  several  sections  was  found  to  be  practically 
constant  for  each  rope,  and  varied  between  O.GloP  and 
0.65d2 — the  mean  value  being  0.63tf,  from  which  we 
obtain  the  ratio 


Area  of  section  of  rope  0.636^* 


Area  of  circumscribing  circle  "    0.7854^a 


=  0.81; 


that   is,  the   actual  area  of  a  3-strand  rope  equals  eight 
tenths  of  the  area  of  the  circumscribing  circle. 


FIG 


CROSS-SECTION  OF  ROPE. 


A  print  obtained  with  a  plaster  cast,  showing  the  distor- 
tion of  the  strands  from  a  true  circle,  is  reproduced  in 
Fig.  40.  Fig.  41  is  a  similar  print,  showing  the  relation 
between  the  actual  area  of  the  cross-section  and  the  area 
included  between  circular  strands  and  the  tangents  joining 
them.  It  will  be  noted  that  the  excess  of  area  outside  of 


94  ROPE-DRIVING. 

the  lines  drawn  tangent  to  the  inner  circles  is  but  slightly 
greater  than  that  between  the  tangents  and  the  inner 
circles. 

An  inspection  and  tabulation  of  the  results  of  numerous 
tests  on  manilla  rope*  shows  that  the  strength  per  square 
inch  increases  as  the  diameter  of  the  rope  decreases. 
Formulas  for  the  strength  of  rope  based  upon  the  circum- 
ference and  a  constant  multiplier — as,  for  instance,  S  = 
800ca,  where  S  is  the  breaking  strength  and  c  the  circum- 
ference— must  be  regarded  as  giving  only  an  average  value 
for  diameters  approximating  those  experimented  upon. 
As  the  strength  of  good  manilla  rope  varies  from  10,000 
pounds  per  square  inch  for  a  2-inch  rope  to  over  12,000 
pounds  for  a  half-inch  rope,  it  can  be  seen  that  a  more 
accurate  determination  may  be  made  if  a  variable  is  used 
in  the  formula.  With  the  above  assumption  of  area  ratios, 
obtained  by  trial,  the  following  expression  has  been  de- 
duced, which  will  give  a  very  fair  value  of  the  breaking 
strength  for  new  manilla  ropes:  ^  =  WQd*x,  in  which  x 
is  a  variable  depending  upon  the  diameter  of  rope;  for 
manilla  rope  we  may  assume  the  empirical  value 
x  —  81  —  9<tf,  where  d  is  the  diameter  in  inches. 

The  ultimate  strength  per  square  inch  of  actual  section 
will  then  be 

S  = 


From  these  formulas  Table  IV  has  been  computed : 
The  above  values  are  for  new  manilla  ropes  made  of 
selected  stock;  ropes  that  are  greasy  or  wet  will  be  reduced 

*  Major  Parker's  report  of  tests  made  at  Watertown  Arsenal, 
1885.  Ex.  Doc.,  No.  36.  Experiments  of  M.  Doboul  in  "Bulletin 
de  la  Societe  d'Encouragemcnt  des  Arts,"  Paris,  1888.  Riehle  Bros,' 
laboratory  tests  ;  and  otheir , 


HOPE-DRIVING.  95 

TABLE  IV.— STRENGTH  OF  MANILLA  TRANSMISSION-ROPES. 
Dia.  of  rope  =  d.         Breaking  Strength,        strength  per  square  inch,  8. 

%  ' 1,900  12,200 

£  2,900  11,950 

£  4,100  11,750 

i  5,500  11,525 

1  7,100  11,350 
H  8,800  11,125 
li  10,900  10,975 
1|  15,000  10,625 
If  19,800  10,300 

2  25,100  10,000 

in  strength  from  20  to  30  per  cent,  but  when  ropes  are 
used  for  transmission  of  power  the  lubrication  of  the 
fibres  is  of  more  importance  than  the  actual  breaking 
strength  of  the  rope,  as  in  any  case  the  apparent  working 
strain,  as  calculated  from  the  power  transmitted  and  the 
speed  of  the  rope,  should  not  be  taken  greater  than  5  per 
cent  of  the  ultimate  strength;  in  many  cases  not  more 
than  2  per  cent  is  used. 

If  we  wished  to  obtain  the  working  strength  in 
any  given  case  we  would  usually  divide  the  breaking 
strength  by  an  assumed  apparent  factor  of  safety,  but 
for  a  flying  rope,  in  addition  to  this,  it  is  necessary 
to  provide  a  factor  of  wear;  moreover,  the  actual  strains 
are  generally  much  greater  than  the  normal  as  calcu- 
lated from  the  power  transmitted,  due  to  vibrations  in 
running,  imperfect  tension  mechanisms,  defects  in  con- 
struction, and  other  causes.  As  the  strength  at  the  splice 
is  only  70  to  75  per  cent  of  the  strength  of  the  rope,  the 
actual  margin  for  wear  and  unknown  strains  is  not  as  large 
as  would  at  first  seem  apparent.  In  the  first  place,  the 
strength  of  a  lubricated  rope  is  weakened  about  25  per 
cent  by  the  grease,  then  25  per  cent  more  must  be  deducted 
for  the  strength  of  the  splice;  this  leaves  only  about  50 
per  cent  of  the  original  strength  of  the  rope  which  is 


96 


ROPE-DRIVING. 


I 


8      a 

bfl 


*  a    to   a    s 

S  E      B      B     ^ 


GO    s-     02    CO 


M 


51 


a  ,     0  .73 

a-r^a 

1   61 


-T=  0  &  0       S3   c  "S  53 

a*^Sr^-c±^'>fl 

I6a8 


(??C<iC50O(??GQOO(?^O-'JC-7OTHO7<MC5 
1—  l-i—ti—  !C?~H-i—  lOir-l-!—  •  i—  I  1C  T-  (i—  1  i-H  i—  l 


O 
fM 


CO        JO  (N  JO        O3  JO  JO  1O  O  O  «O  CO        O3 

COCOCOi— iCOOrHCQC3COi>QO!OCOO5COO 

r-i  O)  O?  (75  03  W  <M  C?  <tt'  (M*  T-i  rH  T-!  ^H  rH  rH  (^ 


•  O  OS  O  00  O?  CO  O 


fctf 


CO  <M  CO  CO  <W  C5  CO 


O  ^5  O  O  O  O  >O  O  1O  iO  ' 


ROPE-DRIVING 


98  EOPE-DRIVIKG. 

available  for  transmission  of  power.  To  allow  for  possible 
imperfections  in  the  rope,  due  to  its  manufacture  or  the 
material  used,  we  must  allow  a  further  reduction  of,  say, 
one  fifth  (that  is,  10  per  cent  of  the  original  strength  of  an 
unlubricated,  unspliced  rope);  thus  we  have  only  40  per 
cent  of  the  original  breaking  strength,  which  we  may  con- 
sider as  the  actual  working  strength.  Allowing  10  per 
of  this  as  the  apparent  working  strain  (equal  4  per  cent  of 
original  strength),  we  obtain  36  per  cent  as  the  actual 
margin  for  wear  and  unknown  stresses  which  may  be  set 
up  in  the  rope;  that  is,  we  have  practically  only  an  actual 
factor  of  safety  of  ten,  used  in  its  ordinary  acceptation, 
instead  of  twenty-five,  as  would  appear  from  the  low  work- 
ing stress. 

The  working  strain  in  executed  rope  transmissions  will 
be  found  to  vary  considerably,  as  shown  in  Table  V  ; 
but  plants  which  have  been  successful,  as  well  as 
those  in  which  the  wear  of  the  rope  was  destructive,  indi- 
cate that  200^ a  pounds  is  an  economical  working  strain.* 

As  this  value  is  such  a  small  percentage  of  the  breaking 
strength,  it  is  unnecessary  to  use  a  coefficient  varying  with 
the  diameter  of  rope,  as  the  difference  is  not  worth  con- 
sidering. •  From  data  furnished  by  the  Messrs.  Pearce 
Brothers,  of  Dundee,  who  have  erected  rope-belting  exten- 
sively, Prof.  Unwin  shows  that  in  different  cases  in  practice 
the  driving  force,  or  difference  in  tension,  on  the  two  por- 
tions of  the  rope  is  equal  to  75  to  80  d*  pounds;  that  is, 

Tl  —  T^  =  P  =  75  to  80  d\ 

where  T,  =  tension  on  driving  side; 
T9  =  tension  on  slack  side; 
P  =  driving  force. 

*  See  paper  by  C.   W.    Hunt,   Trans.    A.  S.  M.  E.,   vol.   xn. 
page  230. 


KOPE-DBIYING.  99 

T 

It  is  probable  that  the  value  of  the  ratio  ~  varies  from 

!-£  to  2-J  iu  ordinary  practice,  depending  upon  the  speed  of 
the  rope,  the  coefficient  of  friction,  and  the  arc  of  contact 
between  rope  and  pulley. 

T 

Unwin  assumes  -£  =  1.2  when  the  belt  embraces  0.4  of 

the  circumference  of  the  smaller  pulley;  hence  the  greatest 
tension  would  be 

Tl  =  1.2 P  =  90tfa  to  96d2. 

This  is  based  upon  a  coefficient  of  friction  =  0.7  for  a  45° 
groove,  which  we  believe  to  be  greatly  in  excess  of  its  aver- 
age value  for  running  ropes.  Moreover,  the  rope  velocity 
was  not  considered  in  determining  the  above  ratio;  at 
speeds  over  2000  feet  per  minute  the  influence  of  cen- 
trifugal force  cannot  be  neglected,  as  it  produces  a  very 
considerable  force  in  the  rope,  and  for  these  reasons  the 

T 

value  of  the  ratio  -~  will  be  greater  in  average  practice 

than  1.2. 

If  we  assume  a  speed  of  4000  feet  per  minute,  the  value 
of  this  ratio  for  greasy  ropes  may  be  taken  equal  to  2,  from 
which  there  is  obtained  Tl  =  2  x  75d2  to  8Qd*  =  150rf*  to 
IQOd'2 — a  value  somewhat  less  than  that  which  we  have  as- 
sumed as  a  suitable  working  strain,  viz.,  200tP  pounds.  In 
a  recent  communication  from  Messrs.  Combe,  Barbour  & 
Combe  of  Belfast,  who  have  been  engaged  in  furnishing 
rope  transmission  for  thirty  years,  it  is  stated  that  their 
basis  of  calculating  the  horse-power  is  to  assume  that  u 
rope  5^  inches  circumference  (If  inches  diameter)  work- 
ing on  a  four-foot  pulley  going  100  revolutions  per  minute 
will  drive  8  h.  p.  under  medium  circumstances;  and  by 
•'•'  medium  circumstances"  they  mean  "  that  the  ropes  must 
work  at  a  distance  of  at  least  20  feet  from  centre  of  shafts 


100  HOPE-DRIVING. 

and  at  a  less  inclination  than  40°  from  the  horizontal,  at  a 
speed  not  under  2000  feet  per  minute.  Should  the  ropes  be 
working  vertically  or  at  an  angle  greater  than  45°,  instead 
of  taking  8  h.  p.  as  the  basis,  you  should  take  7  or  6  respec- 
tively, according  as  the  conditions  grow  worse  and  worse. 
On  the  other  hand,  should  the  ropes  work  horizontally 
and  at  a  greater  distance  than  20  feet  from  centre  to  centre, 
and  with  a  speed  up  to  3600  feet  per  minute,  and  the  pul- 
leys be  fairly  large,  say  from  5  to  7  feet  in  diameter,  you 
may  take  10  instead  of  8  as  the  basis.''  From  these  consid- 
erations the  working  strain  may  readily  be  determined.  If, 

T 

as  before,  we  take  — '  =  2,  we  find  that  under  average  con- 
ditions the  allowable  working  strain  will  be  T^  =  135d3 
pounds.  Under  more  favorable  conditions  and  a  higher 

T 
velocity  -=j  will  be  greater  and  T^  will  approach  200*1*. 

Although  we  have  assumed  the  normal  working  load  not 
to  exceed  200d'2  pounds,  this  must  be  considered  as  the 
economical  load  for  the  lasting  qualities  of  the  rope;  in 
many  cases,  however,  the  first  cost,  the  more  convenient 
adaptation  of  smaller  ropes,  and  the  use  and  lesser  cost  of 
smaller  pulleys  outweigh  the  greater  economy  obtained  by 
the  larger  ropes,  and  loads  are  carried  far  in  excess  of  that 
given.  By  the  use  of  a  greater  number  of  wraps  the  work- 
ing load  on  each  will  be  reduced;  but  this  adds  to  the  first 
cost  of  the  plant,  and  many  concerns  prefer  to  put  in  a  new 
rope  every  year  or  two  rather  than  put  in  more  or  larger 
ropes  every  six  or  eight  years. 

The  ropes  most  commonly  found  in  use  vary  from  f  inch 
to  2  inches  in  diameter,  although  other  sizes  are  frequently 
employed;  in  one  case,  cited  by  Mr.  T.  S.  Miller,*  a  rope 

*  Trans.  A.  S.  M.  E.  1891. 


ROPE-DRIVIM.  101 

no  larger  than  -f^  inch  diameter  is  used  very  satisfactorily 
to  transmit  20  h.  p. 

The  largest  rope  in  use  for  this  purpose,  so  far  as  the 
writer  is  aware,  is  3|  inches  in  diameter:  it  is  of  cotton 
(Lambeth),  and  is  used  to  drive  the  cable  drums  in  the 
Washington  Street  power  station  of  the  North  Chicago 
Street  Railway  Company. 

In  England  and  Germany  ropes  smaller  than  1  inch 
in  diameter  are  seldom  employed  except  for  machine 
driving ;  in  ordinary  cases  the  sizes  most  frequently 
adopted  vary  from  1-J  to  2  inches  in  diameter. 

In  main  drives  where  a  number  of  ropes  are  used,  the 
tendency,  as  judged  from  recent  practice,  seems  to  favor  a 
diameter  about  If  or  If  inches. 

With  a  given  velocity  and  working  tension  the  weight  of 
rope  required  for  transmitting  a  given  horse-power  will  be 
the  same,  irrespective  of  the  diameter  of  rope ;  the 
smaller  rope  will  require  more  parts,  but  the  weight  will 
be  the  same.  As  we  have  stated,  many  engineers  prefer 
to  use  a  greater  number  of  ropes  over  wide-faced  pulleys; 
but  in  order  to  reduce  the  expense  incident  to  a  large 
number  of  grooves  in  the  pulleys  a  closer  margin  is  allowed 
on  the  smaller  ropes,  and  in  consequence  the  normal  work- 
ing strain  on  each  rope  is  usually  increased  far  in  excess  of 
that  which  would  be  necessary  if  the  same  weight  of  rope 
were  employed  as  for.  the  larger  diameter.  As  a  result  of 
this,  the  small  ropes  are  rapidly  worn  out,  and  frequent 
renewals  become  necessary. 

We  are  aware  tlmt  small  ropes  are  advocated  and  in- 
stalled by  many  engineers,  but  we  believe  this  to  be  wrong 
both  in  principle  and  practice.  In  rope-driving  the  first 
cost  and  erection  of  small  driving-pulleys  may  influence  a 
designer  to  use  small  ropes;  but  as  ropes  are  sold  by  the 
pound,  and  as  the  weight  necessary  to  transmit  a  given 
power  should  be  the  same,  irrespective  of  the  diameter,  it 


102  ROPE-DRIVING. 

is  evident  that  the  first  cost  of  the  rope  itself  will  be  the 
same,  whether  large  or  small  ropes  are  employed.  More- 
over, if  the  pulley  is  proportioned  to  the  size  of  rope  in 
each  case,  the  smaller  rope  will  last  only  about  one  third  as 
long  as  a  rope  twice  its  size,  under  similar  conditions. 


HOPE-DRIVING.  103 


CHAPTER  VII. 

HOWEVER  desirable  it  may  be  to  use  a  given  diameter  of 
rope,  the  conditions  of  the  problem  frequently  prohibit  the 
employment  of  such  ropes,  and  the  designer  must  deter- 
mine whether  he  shall  use  a  smaller  diameter  or  a  different 
material  or  method  of  transmission. 

If  we  assume  that  there  is  a  minimum  diameter  of  pulley 
which  may  be  safely  used  for  any  given  diameter  and  speed 
of  rope,  it  will  be  evident  that  the  number  of  revolutions 
of  the  pulley  imposes  conditions  which  limit  the  choice  of 
rope  diameter. 

Thus  if  the  maximum  speed  of  rope  be  taken  at  5000 
feet  per  minute  for  a  permanent  installation,  in  which  the 
working  load  is  200^2  pounds,  and  the  least  diameter  of 
pulley  D  =  d™  ( V  F)  + 12"  (page  179),  then  the  greatest 
number  of  revolutions  which  can  be  obtained  under  these 
conditions  with  a  1-inch  manilla  rope  will  be  550. 

If  the  least  diameter  of  pulley  for  cotton  ropes  be  taken 
equal  to  O.&Z),  then  the  greatest  speed  will  be  approxi- 
mately 700  revolutions  per  minute  for  the  same  diameter 
of  rope.  If  a  greater  rotative  speed  be  desired,  it  is  evident 
that  a  smaller  rope  must  be  used. 

In  any  case,  the  greatest  number  of  revolutions  which 
may  be  obtained  without  excessive  wear  for  a  given  diame- 
ter of  rope  will  be  found  in  Table  VI,  which  has  been  de- 
termined from  the  formula  N=  — =:,  in  which 

TtJJ 

XT  —  revolutions  per  minute  of  smaller  pulley; 
V  =  velocity  of  rope  in  feet  per  minute; 
D  =  least  permissible  diameter  of  pulley. 


104 


ROPE-DRIVING. 


For  machine-driving  greater  speeds  are  obtained  by  the 
use  of  smaller  ropes  than  those  given  in  the  table;  but  it 
is  not  advisable  in  most  cases  to  use  a  rope  less  than  f  inch 
diameter  for  general  transmissions. 

TABLE  VI.— GREATEST  REVOLUTIONS  PER  MINUTE  FOR  GIVEN 
DIAMETER  OF  ROPE. 


Diameter 
of 
Rope. 

Maximum  Revolutions  per  Minute  of  Smaller  Pulley 
corresponding  to  a  Linear  Velocity  of 
5000  Feet  per  Minute. 

Manilla. 

Cotton. 

i 

ii 

H 

i4 

710 
550 
430 
350 

280 
240 

890 
670 
530 
440 
350 
290 

The  wear  of  a  rope  is  both  internal  and  external.  As  we 
have  previously  noted,  the  internal  wear  is  due  to  the  bend- 
ing of  the  fibres  and  their  sliding  upon  one  another,  which 
produces  a  grinding  action, — very  much  increased  when  the 
strands  are  not  lubricated  or  when  a  hard  twist  is  given  to 
the  rope,  thus  preventing  by  the  greater  compression  of  the 
fibres  upon  one  another  that  freedom  of  action  which  is  so 
essential  to  the  life  of  the  rope.  It  is  evident  that  a  similar 
compression  of  the  fibres  will  occur  when  a  rope  under  ten- 
sion is  wrapped  around  a  pulley:  the  greater  this  tension 
the  greater  also  will  be  the  compression  and  distortion  of 
the  fibres. 

The  external  wear  is  due  to  the  contact  between  the  rope 
and  the  sides  of  the  groove  in  which  it  runs,  and  is  greatly 
increased  when  slip  occurs;  roughness  in  the  groove  also 
increases  the  wear,  and  for  this  reason  the  rim  should  be 
turned  smooth  and  polished,  as  the  outer  fibres,  rubbing 


ROPE-DRIVIKG.  105 

011  a  rough- turned  or  cast  surface,  will  gradually  break, 
fibre  by  fibre,  and  thus  give  the  rope  a  short  life. 

Contact  between  different  ropes,  or  between  a  rope  and 
some  obstructing  surface,  such  as  a  partition  wall,  post,  or 
floor-beam,  is  frequently  the  cause  of  a  large  portion  of  the 
external  wear  of  a  rope :  this  may  be  due  to  faulty  construc- 
tion or  erection;  pulleys  designed  with  too  small  a  pitch 
between  the  grooves,  or  running  out  of  true,  causing  the 
ropes  to  vibrate  and  flap  against  each  other,  or,  as  in  out- 
side work,  a  swaying,  producing  contact,  may  be  set  up  in 
the  ropes,  due  to  the  action  of  the  wind. 

Excessive  swaying  will  also  tend  to  cause  the  rope  to 
jump  its  groove.  In  order  to  prevent  this  and  reduce  the 
side  motion  as  much  as  possible  it  is  often  customary  in 
outdoor  drives  to  place  idlers  for  both  tight  and  slack 
sides  of  the  ropes  so  as  to  guide  each  portion  as  it  enters 
upon  or  leaves  the  groove. 

A  characteristic  uniform  surging  sometimes  occurs  in 
flying  ropes  due  to  the  harmonic  vibration  which  is  set  up 
when  the  speed  and  distance  between  shafts  bears  a  cer- 
tain relation  to  the  tension  in  the  ropes.  Cases  of  exces- 
sive vibration  due  to  this  cause  have  been  remedied  by 
slightly  increasing  or  decreasing  the  speed  of  the  ropes. 
Where  such  vibration  causes  the  rope  to  beat  against  an 
obstruction,  as  a  floor  or  ceiling,  the  external  wear  is  of 
course  increased. 

With  the  same  total  stress  in  a  rope,  it  may  be  assumed 
that  the  wear,  both  internal  and  external,  increases  directly 
with  the  number  of  flexures,  the  slip,  and  the  surface  in 
contact;  and  also,  that  a  reverse  bending  is  more  injurious 
to  the  rope  than  single  bends  in  a  constant  direction. 
For  a  given  speed  the  number  of  flexures  and  the  actual 
surface  in  contact  with  the  pulleys  will  decrease  as  the 
distance  between  centres  increases,  and  hence  the  wear 
will  vary  inversely  with  the  distance  between  centres  of 


106  HOPE-DRIVING. 

pulleys,  but  it  must  be  noted  that  with  imperfect  con- 
struction an  increased  distance  between  shafts  will  favor 
swaying  and  rubbing  of  the  ropes  against  each  other  and 
the  edges  of  the  grooves. 

The  number  of  flexures  and  the  surface  in  contact  will 
evidently  increase  directly  as  the  velocity,  and  therefore 
the  wear  may  be  assumed  to  vary  directly  as  the  velocity  of 
the  rope.  If  we  assume  that  two  IJ-inch  ropes  are  neces- 
sary to  transmit  a  given  horse-power,  it  will  require  eight 
f-inch  ropes  to  transmit  the  same  power  at  the  same  speed 
and  tension  per  square  inch  of  section.  If  suitable  pul- 
leys are  used  in  each  case  the  wear  will  be  considerably 
greater  with  the  smaller  ropes.  For  the  total  external 
surface  of  the  eight  ropes  in  contact  with  the  pulley  each 
revolution  is  twice  as  great  as  that  obtained  with  the  1J-- 
inch  ropes;  moreover,  the  distance  between  centres  of 
shafts  will  generally  be  considerably  less  with  the  smaller 
ropes,  and  as  the  number  of  revolutions  of  the  smaller 
pulleys  should  be  more  than  twice  as  great  for  the  same 
speed  of  rope  (since  the  pulleys  are  less  than  half  the  size 
of  those  used  for  the  larger  rope;  see  page  180),  the  number 
of  flexures  and  the  wear  of  the  rope  on  the  pulleys  will  be 
greater  with  the  f-inch  rope  ;  for  not  only  is  each  rope 
bent  more  than  twice  as  many  times  per  minute,  thus  pro- 
ducing eight  times  the  bending  in  the  smaller  ropes,  but, 
as  the  slip  is  independent  of  the  diameter  of  the  rope,  it 
will  be  evident  for  the  same  proportional  stress  and  arc  of 
contact  that  the  slip  will  be  four  times  greater  in  the  case 
of  the  smaller  ropes,  even  if  we  neglect  the  differential 
driving  effect  which  may  be  assumed  to  increase  with  the 
number  of  ropes. 

Thus  it  will  be  seen  that  under  similar  conditions  and 
proportional  stress,  we  should  expect  the  smaller  rope  to 
wear  out  more  than  twice  as  fast  as  one  double  its  size; 
and  when  the  stress  is  proportionally  greater  in  the  smaller 


ROPE-DRIVING.  107 

rope,  as  we  ordinarily  find  it,  the  wear  will  be  still 
greater. 

These  conclusions  are  borne  out  in  practice,  for  in  trans- 
missions using  small  ropes,  f-inch  in  diameter  and  under, 
the  life  of  manilla  ropes  is  usually  only  from  six  to  twelve 
months;  in  many  cases  such  ropes  will  last  only  three 
months,  although  others  have  been  in  active  service  for 
periods  varying  from  one  to  two  years. 

On  the  other  hand,  the  larger-sized  ropes,  one  inch  to 
two  inches  in  diameter,  will  last  from  two  to  six  years,  and 
under  favorable  conditions  large  ropes  have  lasted  eight 
and  even  ten  years. 

The  same  is  true  regarding  cotton  ropes. 

The  comparatively  short  life  of  small  ropes  used  in 
machine-driving  has  led  to  the  belief  that  cotton  ropes 
wear  out  rapidly;  but  such  an  impression,  at  least  regard- 
ing the  larger  sizes,  is  altogether  erroneous,  as  these  ropes 
when  properly  put  on  and  cared  for  will  give  good  service 
for  ten  or  twelve  years. 

Messrs.  John  Musgrave  &  Sons,  Bolton,  Eng.,  who  have 
had  a  large  experience  with  rope-driving,  state  that  some 
of  their  ropes  have  been  in  use  for  seventeen  years  and  were 
still  in  good  order. 

The  life  of  a  rope,  whether  of  manilla  or  cotton,  will  de- 
pend altogether  upon  the  work  it  has  to  do  and  the  atten- 
tion it  receives. 

Two  ropes  cut  from  the  same  coil  can  be  put  to  work  on 
different  drives,  and  one  will  last  only  six  months  while 
the  other  will  be  in  good  condition  after  continual  service 
for  ten  years. 

From  an  investigation  of  numerous  examples  of  rope- 
driving  under  a  variety  of  existing  conditions  the  writer 
is  led  to  believe  that  ropes  less  than  f  inch  diameter 
should  not  be  used  if  it  is  at  all  practicable  to  employ  tke 
larger  sizes,  and  that  ropes  one  mch  in  diameter  and  over 


108  ROPE-DRIVING. 

are  to  be  preferred  where  it  is  possible  to  use  the  larger 
pulleys  which  are  necessary  for  such  ropes. 

With  larger  ropes  the  wear  is  not  only  much  less,  but, 
where  the  nsual  multiple-wrap  system  is  used,  when  a, 
number  of  yarns  give  way  the  rope  does  not  part  at  once 
if  subjected  to  a  greater  or  sudden  tension,  but  may  run 
until  a  convenient  opportunity  offers  to  shut  down;  where- 
as with  the  small  rope,  having  a  greater  number  of  wraps, 
when  a  strand  or  a  number  of  yarns  give  way,  any  in- 
creased stress  due  to  additional  load  or  imperfections  in 
the  system  is  liable  to  still  further  rupture  the  yarns  in  the 
weaker  rope,  and  a  sudden  break  or  pulling  out  occurs. 

Besides  the  greater  life  of  the  rope,  and  consequent  less 
cost,  the  saving  of  time  on  account  of  fewer  breakdowns 
and  stoppages  is  a  factor  worth  considering;  and  although 
this  feature  does  not  cut  as  much  of  a  figure  with  rope- 
driving  as  it  does  with  factory  belting,  yet  it  is  of  such 
importance  that  many  men  would  rather  use  some  other 
form  of  transmission  than  suffer  the  annoyance  incident  to 
respl  icing  a  broken  rope  every  few  months.  While  the 
time'lost  in  repairing  the  rope  or  in  laying  down  a  new  one 
may  not  be  great  in  itself,  the  stoppage  of  a  department  in 
a  busy  season  may  prove  to  be  a  serious  loss.  Where 
foundries  and  isolated  shops  have  been  driven  by  small 
ropes  this  has  happened  so  frequently  that  the  ropes  have 
been  taken  out  and  replaced  with  shafting,  or  other  more 
expensive  method  of  transmission. 

In  order  to  avoid  any  serious  loss  of  time  or  inconven- 
ience due  to  a  sudden  rupture  two  independent  ropes 
should  be  used,  each  having  its  own  tension  sheave  and 
weight.  This  does  not  involve  any  more  wraps  than  would 
otherwise  be  used  for  a  single  wind,  for  the  normal  work- 
ing stress  is  in  any  case  so  much  less  than  the  actual 
strength  that  for  a  temporary  run  of  a  few  hours,  or  even 
days,  one  rope  could  readily  carry  double  its  working  load 


ROPE-DRIVING.  109 

in  case  the  other  should  give  out.  When  two  independent 
ropes  are  thus  used  they  may  be  wound  separately,  each 
wrap  occupying  successive  grooves  on  the  pulleys  ;  or, 
which  is  more  frequently  the  case,  the  ropes  may  be  wound 
in  parallel,  thus  bringing  each  rope  in  alternate  grooves- 
in  either  case  it  is  preferable  to  use  an  independent 
tightener,  although  a  single-tension  carriage,  provided 
with  two  sheaves,  may  be  used;  but  with  this  latter,  owing 
to  local  causes  and  the  difficulty  of  splicing  both  ropes  of 
an  equal  length,  the  load  is  not  as  well  distributed  between 
the  two  ropes. 

In  subsequent  considerations  of  the  driving  power  of 
ropes  the  relation  between  the  ultimate  strength,  weight 
per  foot  of  length,  normal  working  strain,  and  the  diame- 
ter of  rope  will  be  represented  by  the  following  equations 
which  have  been  determined  for  manilla  transmission  rope  : 

Let   d  =  diameter  of  rope  in  inches; 

w  =  weight  of  rope  in  pounds  per  foot; 
St  =  breaking  strain  in  pounds; 
t  =  normal  working  strain  in  pounds; 
x  =  an  empirical  coefficient. 
Then  w  ='.0.3216^; 
8  = 


x  =  81  -  9& 

The  weight  w  per  foot  of  length  varies  considerably  m 
different  makes  of  rope,  depending  upon  the  amount  of 
twist  and  the  foreign  matters  in  the  rope. 

It  is  well  known  that  much  of  the  cheaper  manilla  rope 

*  On  account  of  the  relatively  great  difference  between  &  and  t,  it 
is  not  thought  advisable  to  consider  the  increase  in  strength  of  the 
smaller  ropes,  as  in  any  case  the  difference  would  be  very  slight,  and, 
moreover,  it  must  be  noted  that  t  is  the  normal  estimated  strain,  and 
may  vary  considerably  from  the  actual  strain. 


110 


ROPE-DRIVING. 


on  the  market  is  largely  adulterated  with  weighing  mate- 
rial, such  as  gelatine  size,  French  clay,  and  white  lead. 
Thus  some  manilla  ropes  will  weigh  not  more  than  0.26d~ 
pounds  when  dry  and  very  loosely  twisted;  in  other  cases 
the  weight  will  be  as  much  as  0.46<:f  pounds  per  foot  of 
length.  The  value  we  have  given,  viz.,  0.32d2  pounds,  cor- 
responds very  closely  to  the  average  weight  of  good  quality 
lubricated  transmission  ropes. 

Cotton  ropes  are  about  twenty  per  cent  lighter  for  equal 
diameters,  and  will  vary  from  0.20d2  to  0,29d2  pound  per 
foot.  In  the  following  table  (VII)  0.32d2  has  been  used 
for  manilla  and  Q.ZGd2  for  cotton  ropes. 

TABLE  VII.— WEIGHT  OF  ROPES 


Weight  in  Pounds  per  Foot. 

Diameter 

of 

Rope. 

Manilla. 
w  =  0.32d». 

Cotton. 
w  =  0.26d*. 

f 

0.18 

0.15 

.32 

.26 

H 

.50 

.40 

H 

.72 

.58 

If 

.98 

.79 

2 

1.28 

1.04 

ROPE-DBIVItfG.  Ill 


CHAPTEE  VIII. 

IN  determining  the  horse-power  which  a  rope  will  trans- 
mit under  given  conditions  the  centrifugal  force  due  to  the 
velocity  and  weight  of  rope  is  an  important  factor,  and  its 
influence  should  be  considered  in  all  cases  where  the 
speed  is  greater  than  2000  feet  per  minute;  for  at  high  ve- 
locities this  force  diminishes  the  pressure  exerted  between 
the  rope  and  the  circumference  of  the  pulley,  thus  reducing 
the  friction  between  rope  and  pulley.  When,  therefore,  a 
rope  has  to  transmit  a  given  force,  P,  it  must  be  subjected 
to  a  greater  tension  the  greater  the  centrifugal  force  F0. 
At  a  speed  of  about  90  feet  per  second  the  centrifugal 
force  increases  faster  than  the  power  from  increased  veloc- 
ity of  the  rope,  and  at  140  feet  per  second  this  force  equals 
the  assumed  allowable  back  tension  in  the  rope;  and  since 
the  transmitting  force  is  equal  to  the  difference  in  tension 
in  the  two  parts  of  the  rope,  it  will  be  seen  that  no  power 
will  be  transmitted  at  this  speed  unless  the  assumed  allow- 
able tension  be  exceeded. 

It  is  evident  that  for  a  given  total  tension  the  less  back 
tension  required  to  prevent  the  slip  of  the  rope  on  the 
pulley  the  greater  will  be  the  power  transmitted  at  a  given 
speed. 

The  determination  of  this  back  tension  is,  however,  at- 
tended with  a  degree  of  uncertainty,  as  there  are  no  con- 
clusive experiments  which  give  reliable  data  for  its  calcu- 
lation; the  coefficient  of  friction,  0,  as  stated  by  various 
authorities,  varying  all  the  way  from  0.075  up  to  0.88.* 

*  The  probable  reasou  for  such  widely  divergent  values  lies  in  the 
fact  that  the  coefficient  of  friction  varies  with  the  percentage  of 
slip,  and  those  tests  made  with  very  little  slip  would  show  a  small 


.112  ROPE-DRIVING. 

Reuleaux  quotes  the  experiments  of  Leloutre  and  others 
as  indicating  a  value  of  0.075  for  cylindrical  pulleys  with 
new  hemp  rope,  0.088  for  semicircular  grooves,  and  0.15 
for  a  wedge  groove  of  60°. 

Experiments  by  the  Messrs.  Pearce  Brothers,  of  Dundee, 
give  a  value  of  0  equal  to  0. 57  to  0.88  for  ropes  on  ungreased 
pulleys,  and  <p  —  0.38  to  0.41  when  the  pulleys  are  greased. 

Unwin  states  that  the  coefficient  of  friction  for  ropes  on 
a  flat-metal  pulley  is  equal  to  0.28,  from  which  the  actual 
coefficient  for  a  grooved  pulley  is  obtained  by  multiplying 
0.28  by  the  cosec.  of  half  the  angle  of  the  groove.  For  an 
angle  of  45°  this  would  give  0  =  0.72.  These  latter  values 
are  probably  very  much  higher  than  is  ordinarily  found  in 
actual  practice  with  well-lubricated  ropes  and  moderate 
slip.  From  a  consideration  of  the  above  and  various  other 
experiments,  and  the  conditions  under  which  they  were 
carried  out,  it  would  appear  that  for  ropes  which  are  partly 
worn  and  sufficiently  greased  to  wear  well  with  a  low  per- 
centage of  slip  a  value  of  0.12  for  a  flat-surfaced,  smooth- 
metal  pulley  will  approach  very  closely  to  those  conditions 
which  obtain  in  average  practice,  from  which  the  following 
working  coefficients  are  deduced: 


0  =  0.12  cos 

/Angle  of  groove^ 

-c  v 

2 

/ 

Angle  of  groove                  . 

30° 
0.46 

35° 

.40 

40° 
.35 

45° 
.31 

50° 

.28 

55° 
.26 

60' 
.24  { 

Coefficient  of  friction,  0... 

Besides  varying  with  the  angle  of  groove,  as  shown,  the 

coefficient;  on  the  other  hand,  it  is  safe  to  assume  that  the  larger 
values  were  obtained  under  conditions  in  which  the  slip  was  greatly 
increased.  Varying  atmospheric  conditions  and  different  degrees  of 
lubrication  are  also  largely  responsible  for  these  divergent  results. 

See  paper  by  Prof.  Lanza  on  ""Friction  of  Leather  Belting,"  in 
Trans.  A.  S.  M.E.,  vol.  vn.  p.  347;  also  "Experiments  on  Power 
Transmitted  by  Belting,"  by  Wilfred  Lewis,  in  Trans.  A.  S,  M.  E. 
vol.  vn.  p.  549. 


ROPE-DRIVING.  113 

coefficient  of  friction  is  affected  by  the  condition  of  the 
rope,  and  for  dry  ropes  0  may  be  taken  somewhat  greater 
than  the  above  value;  0  will  also  be  increased  with  an  in- 
creased percentage  of  slip. 

If  the  arc  of  contact  on  smaller  pulley,  the  coefficient 
of  friction  between  rope  and  sheave,  and  the  total  tension 
in  the  rope  be  known,  the  tension  on  slack  side  of  the 
pulley,  and  hence  the  horse-power  transmitted,  can  be 
readily  determined  from  the  following  considerations:* 

Assuming,  as  before,  that  the  driving  force  P  is  equal 
to  the  difference  in  tension  2\  on  the  driving  side  of  the 
rope  and  Ty  on  the  driven  side,  and  noting  that  the  driv- 
ing force  must  equal  the  friction  F  between  the  surfaces, 
we  obtain 

Tl  -  T^  =  P  =  F. 

The  friction  F  depends  upon  the  arc  of  contact  a  between 
the  rope  and  its  sheave,  the  coefficient  of  friction  0,  and 
upon  the  centrifugal  force  F0  set  up  in  the  rope,  due  to 
its  velocity  and  weight;  it  is,  however,  independent  of  the 
diameter  of  pulley.  To  determine  the  values  of  F9  Tlf 
and  Ty  it  will  be  necessary  to  assume  a  given  tension  in 
the  rope;  .also  its  speed  and  weight,  coefficient  of  friction, 
and  arc  of  contact. 

Let/  =  friction  between  element  of  rope  and  pulley; 
g  =  acceleration  due  to  gravity  =  32.16; 
p  =  normal  pressure  exerted  by  element  on  pulley; 
t  =  allowable  working  tension  =  200^2  pounds; 
v  =  velocity  of  rope  in  feet  per  second; 
w  =  weight   of    rope    per    unit    length    and    area 
=  0.3216d2  pounds  per  foot; 


z  =  abbreviation  for  — -: 


*Weisbach,  vol.  in.  p.  254.    See  also  Eeuleaux. 


114  ROPE-DRIVING. 

A  =  area  of  cross-section  of  rope; 

F0  =  centrifugal  force  due  to  speed  and  weight  of 

rope; 

P  =  driving  force  =  Tl  —  T^ 
R  =  radius  of  pulley; 
T  =  tension  in  rope  at  any  point; 
TI  =  tension  in  rope  on  tight  side; 
T^  =  tension  in  rope  on  slack  side; 
a  =  least  arc  of  contact  between  rope  and  pulley 

circular  measure  —  0.0175  X  arc  in  degrees; 
0  =  coefficient  of  friction. 

If  in  Fig.  42  T  is  the  tension  in  the  rope  at  any  point 
D,  then  the  tension  at  the  point  Ey  whose  distance  from 
D  is  ds,  will  be  T  +  dT.  Assuming  /  to  represent  the 


>T+  cfT 


FIG.  42. 

friction  on  the  element  DE,  we  shall  have/=  dT  when 
the  several  forces  are  in  equilibrium.  Since  the  friction 
is  dependent  upon  the  normal  pressure  p  exerted  by  the 
element  upon  the  rim  of  the  pulley,  and  since  this  normal 
pressure  is  diminished  by  the  centrifugal  force  due  to  the 
weight  and  velocity  of  the  element,  we  shall  have 


KOPE-DKIVING.  115 

dT=<l>(p-F.)  ......      (4) 

Now  ^  is  the  resultant  of  the  two  forces  Tand  T  -\-  dT\ 
hence 


which  may  be  assumed  equal  to  Tda  on  account  of  the 
smallness  of  da  and  dT. 

The  centrifugal  force  of  the  element  of  the  rope  is 

Av*  ,          TV* 
F«  =  wARgds  =  WlRgdS> 

and  since  ds  =  Rda,  we  have 

F9  =  w~da. 

t  9 

Substituting  these  values  of  p  and  F0  in  dT  =  <p(p  —  F0), 
we  obtain 


»•-;••/;  - 

For  convenience,  let  z  —~T\  then 

dT 
dT  =  cf>T(l  -  z)da,    or     ^-  =  0(1  -  z)rf*. 

Integrating, 

/•a\  ^r         /•« 
/       -^  =    /     0(1  -  ^Jrfflf 

•A  70 

log^  =  0(l-^)a;    ...      (6) 


hence 


in  which  e  is  the  base  of  the  hyp  log  =  2.7183;  therefore 


116  HOPE-DRIVING. 

/TT    rn    D  /7'  [~    </>a(l  —  z)~]     .      ^   — — 

J-i       j.  t  —  JT    -  2  i\_e  ^2  — 


(7) 


If  we  assume  -~  =  r,  there  is  obtained 


T, 


r  -  1  ""  T,  -  T,  ~  P' 

T 

These  ratios,  r  and  —  —  -,  Reuleaux  calls  the  friction 

modulus  and  the  stress  modulus,  respectively. 

As  the  common  logarithm  of  e  is  0.43430,  the  value  of 
r  may  be  more  readily  obtained  from  common  log  r  — 
0.4343  0«(1  —  z)  ;  if  the  arc  of  contact  is  given  in  degrees, 
a  =  0.01750-°,  which  gives  the  common  log  ( 


r  =  0.4343  X  .0175a°0(l  -  z)  =  0.0075780a  (1  -  z).    (8) 


As  the  weight  of  a  manilla  rope  one  foot  long  =  0. 
pounds,  the  value  of  z  for  varying  speeds  can  be  deter- 
mined from 


. 

Z  =  —  :-  =  -  —  ......      (9) 

gt  t 

If  now  we  assume  a  constant  working  stress  t  = 
pounds,  then 


In  the  work  which  follows  we  shall  assume  that  the  ten- 
sion Tl  in  the  rope  on  the  tight  side  (driving  tension)  equals 
the  allowable  tension  t. 

From  these  assumptions  the  following  table  (VIII)  of 
the  values  of  1  —  z  has  been  computed  ; 


117 

TABLE  VIII.—  VALUES  OF  1  —  z  FOR  A  WORKING  STRESS  EQUIVA- 
LENT TO  200d2  POUNDS. 


«ueS  of  oy          Values  of 

per  Minute.  per  Minute. 

1000  0.98  5500  0.58 

2000  0.94  6000  0.50 

2500  0.91  6500  0.41 

•  3000  0.87  7000  0.32 

3500  0.83  7500  0.22 

4000  0.78  8000  0.11 

4500  0.72  8500  0.0 

5000  0.65 

It  will  be  seen  from  the  above  that  when  the  velocity  of 
the  rope  is  as  great  as  8500  feet  per  minute,  1  —  2  =  0, 

T 
hence   log  r  =  0   and  -^  —  1;  that  is,   Tl—  T^  =  0,  and 

A 

therefore  no  power  will  be  transmitted  unless  the  assumed 
working  tension  t  be  exceeded. 

In  average  work  the  lesser  arc  of  contact  embraced  by 
the  rope  —  generally  on  the  smaller  pulley  —  will  be  about 
165°,  and  this  value  may  be  assumed  for  approximate  cal- 
culations with  a  working  degree  of  accuracy.  If  the  angle 
in  degrees  is  known,  its  value,  a,  in  circular  measure,  can 
be  obtained  from  Table  IX,  in  which  a  =  0.0175or°. 

TABLE  IX.  —  ANGLE  EMBRACED  BY  ROPE. 

Fraction  of 

Degrees,  Circular  Measure,       Circumference, 

a.  360/a°. 

105  1.83  0.29 

120  2.09  0.33 

135  2.35  0.37 

150  2.62  0.42 

165  2.88  0.46 

180  3.14  0.50 

195  3.43  0.54 

210  3.66  0.58 

240  4.19  0.66 

If  we  now  assume  the  coefficient  of  friction  0  to  be  0.31 
for  a  45°  groove,  we  may  obtain  the  value  of  the  ex- 
pressions 


118  BOPE-DRIVIKCK 


=£  =r  . 

In  order  to  simplify  calculation,  the  following  table  (X) 

contains  values  of  r  and  --  -,  which  will  enable  the  horse- 

power transmitted  by  a  rope  to  be  determined  with  a  degree 
of  accuracy  depending  upon  the  assumption  of  the  coeffi- 
cient of  friction  : 

TABLE  X.—  FRICTION  AND  STRESS  MODULI. 

»$         ^i-£ 

0.1  1.11  10.41 

0.2  1.23  5.40 

0.3  1.35  3.86 

0.4  1.49  3.02 

0.5  1.65  2.54 

0.6  1.82  2.22 

0.7  2.01  1.99 

0.8  2.22  1.82 

0.9  2.46  1.69 

1.0  2.72  1.58 

1.1  3.00  1.50 

1.2  3.32  1.43 

1.3  3.67  1.87 

1.4  4.06  1.33 

1.5  4.48  1.29 

The  following  application  will  show  the  use  of  the  tables. 
Let  it  be  required  to  determine  the  horse-power  transmitted 
by  a  rope  1  inch  in  diameter  running  at  a  velocity  of  4000 
feet  per  minute  over  a  pulley  with  45°  grooves.  Assuming 
an  arc  of  contact  of  165°,  we  find  from  Table  IX  a  =  2.88; 
for  the  required  velocity,  4000  feet  per  minute,  Table  VIII 
gives  0.78  as  the  value  of  1  —  z;  therefore,  assuming  the 
coefficient  of  friction  0  =  .31,  we  obtain 

<pa(l  -z)  =  0.31  X  2.88  X  .78  =  .69. 

T 
From  Table  X  the  value  of  -,  corresponding  to  0.69, 


HOPE-DRIVING. 


119 


is   about   1.99;   and  as  3P  =  200rfa  pounds  =  200   in  the 

T 

present    case,   we    find    P  =  —~  =  100  pounds.     Since 


PV 
33000 


1.99 


there 


100  X  4000 
33000 


12.15,  represented  by  the  ordinate  Im  in  Fig.  43. 
The  loss  due  to  centrifugal  force  may  now  be  obtained 


FT. PER  MINUTE  £lee.W»U 

FIG.  43.— CENTRIFUGAL  EFFECT  IN  ROPES. 

by  assuming  the  latter  reduced  to  zero,  in  which  case  the 
factor  1  —  z  is  equal  to  unity;  therefore  log  r  =  .4340#, 
from  which,  with  the  previous  conditions,  we  obtain  P  = 
120  and  the  corresponding  h.  p.  =  14.5. 

This  value  is  represented  on  the  diagram.  Fig.  43,  by  the 
ordinate  In  :  the  difference  between  In  and  Im  —  mn  will 


120 


ROPE-DRIVING. 


then  be  tke  loss  due  to  the  centrifugal  force  set  up  in  the 
rope  =  14.5  —  12.15  =  2.35  horse-power. 

For  any  special  case  where  the  data  are  known  or  may 


EUe.World 

FIG.  44. — HORSE-POWER  TRANSMITTED  BY  MANILLA  ROPES. 

be  determined,  the  formulas  and  tables  already  given  should 
be  used  to  ascertain  the  horse-power  transmitted,  or  the 
diameter  and  number  of  ropes  required  for  a  certain  work, 
as  the  case  may  be.  For  average  work,  however,  it  will  be 


ROPE-DRIVING. 


found  that  the  assumed  values  of  a  and  0,  previously 
noted,  -will  give  very  satisfactory  results,  and  upon  these 
assumptions  the  writer  has  computed  the  following  table  of 
horse-power  (Table  XI)  for  various-sized  ropes,  running  at 
speeds  from  1000  to  7500  feet  per  minute: 

TABLE  XI. — HORSE-POWER  TRANSMITTED  BY  ROFES. 

Working  Strain  =  200<22  pounds. 
d  =  diameter  of  rope  in  inches. 


Velocity  of 
Rope  in  Feet 
per  Minute. 

Diameter  of  Rope. 

% 

H 

1 

94 

Itt 

m 

2 

1000 

1.24 

2.25 

3.57 

5.59 

8.02 

10.85 

14.20 

2000 

2.70 

3.84 

6.84 

10.68 

15.39 

20.93 

27.36 

2500 

3.30 

4.71 

8.38 

13.10 

18.86 

25.66 

33.54 

3000 

3.83 

5.46 

9.80 

15.39 

21.87 

29.74 

38.88 

3500 

4.30 

6.23 

11.09 

17.33 

24.94 

34.03 

44.35 

4000 

4.74 

6.83 

12.15 

18.98 

27.33 

37.17 

48.59 

4500 

5.01 

7  24 

12.89 

20.15 

29.00 

39.45 

51.57 

5000 

5.20 

7.'47 

13.29 

20.76 

29.89 

40.65 

53.15 

5500 

5.29 

7.60 

13.53 

21,14 

30.43 

41.39 

54.11 

6000 

5.08 

7.32 

13.10 

20.36 

29.32 

39.77 

52.12 

6500 

4.74 

6.83 

12.13 

19.00 

27.34 

87.21 

48.63 

7000 

4.12 

5.93 

10.54 

16.47 

23.72 

32.26 

42.18 

7500 

3.25 

4.67 

8.32 

13.00 

18.73 

25.42 

33.23 

The  graphic  representation  of  these  values,  Fig.  44, 
shows  the  effect  of  centrifugal  force  in  diminishing  the 
power  transmitted  under  an  assumed  working  tension,  and 
would  indicate  that  with  tensions  of  200^2  pounds  the 
speed  should  not  exceed  5500  feet  per  minute.  The  in- 
creasing effect  and  loss  of  power  due  to  centrifugal  force 
in  the  rope  can  also  be  seen  in  the  diagram,  Fig.  43,  which 
represents  the  horse-power  transmitted  by  an  inch  rope 
under  an  assumed  constant  tension  of  200  pounds.  The 
straight  line  AB  shows  the  power  which  would  be  trans- 
mitted if  centrifugal  force  were  neglected,  and  is  obtained 
by  making  z  =  0  in  the  general  equation 


122  ROPE-DRIVItfG. 

log  r  =  0.4343^0(1 -z); 

the  curve  AC  represents  the  power  transmitted  when  cen- 
trifugal force  is  taken  into  account;  and  the  curve  AB 
shows  the  power  absorbed  by  centrifugal  force:  this  latter 
curve  is  obtained  by  subtracting  the  vertical  ordinates  be- 
twejen  the  straight  line  AB  and  the  curve  AC.  By  a  con- 
sidation  of  the  diagram  it  will  be  seen  that  at  speeds 
less  than  2000  feet  per  minute  the  power  absorbed  by 
centrifugal  force  is  very  small,  and  may  be  neglected  as 
far  as  practical  results  are  concerned.  Beyond  this  speed, 
however,  the  loss  from  this  cause  increases  very  rapidly, 
until,  as  previously  shown,  at  a  speed  of  about  8500  feet 
per  minute  the  whole  of  the  allowable  tension  is  absorbed. 

Assuming  that  the  maximum  power  is  transmitted  by  a 
rope  at  a  velocity  of  about  5500  feet  per  minute,  it  is  evi- 
dent that  the  first  cost  of  the  rope  will  be  a  minimum  for  a 
given  power  when  running  at  this  speed.  The  ratio  of  the 
first  cost  of  the  rope  running  at  any  other  speed  may  be 
obtained  by  dividing  the  horse-power  at  5500  per  minute 
by  the  horse-power  at  the  required  speed.* 

Thus,  if  the  first  cost  of  a  IJ-inch  rope  which  will  trans- 
mit 21.14  h.  p.  at  5500  feet  per  minute  be  represented  by 
unity,  the  cost  at  3000  feet  per  minute  will  be 

21.14  _ 
15^9  * 

since  st\ J-inch  rope  running  at  3000  feet  per  minute  will 
transmit  15.39  h.  p. 

The  relative  first  cost  for  a  given  diameter  of  rope  to 
transmit  the  same  horse-power  at  varying  speeds  is  shown 
in  the  accompanying  Table  XII. 

Although  the  first  cost  of  a  rope  to  transmit  a  given  horse- 
power is  a  minimum  for  a  speed  of  about  5500  feet  per 

*  C,  W.  Hunt,  in  Trans.  A.  S.  M.  E.,  vol.  xii. 


ROPE-DRIVING. 


123 


minute,  yet  the  economy  is  not  as  great  as  would  appear  from 
the  foregoing  table,  for  the  effect  of  wear  must  be  considered. 
The  causes  of  wear,  internal  and  external,  have  been  pre- 
viously discussed;  it  will  be  sufficient  to  note  here  that  the 

TABLE  XII. — RELATIVE  FIRST  COST  OF  ROPE-DRIVING. 


Velocity  of  Rope 
in  Feet 
per  Minute. 

Relative  First 
Cost  per 
Horse-power. 

Velocity  of  Rope 
in  Feet 
per  Minute. 

Relative  First 
Cost  per 
Horse-power. 

1,000 

3.78 

4,500 

.05 

2,000 

1.89 

5,000 

:  .02 

2,500 

1.62 

5,500 

.00 

3,000 

1.38 

6,000 

.03 

3,500 

1.22 

6,500 

.12 

4,000 

1.10 

7,000 

.28 

internal  destructive  effect  produced  by  bending  and  distort- 
ing the  fibres  and  the  wear  due  to  external  contact,  slipping, 
or  wedging  in  the  grooves  of  the  pulleys,  may,  within  the 
limits  of  ordinary  practice,  be  considered  as  directly  propor- 
tional to  the  velocity  of  the  rope.  What  this  wear  is  in 
terms  of  the  velocity  there  is  not  sufficient  data  to  deter- 
mine, but  if  the  coefficient  be  represented  by  c,  the  relative 
wear  for  a  given  diameter  may  be  determined  by  multiply- 
ing the  velocity  by  this  coefficient;  that  is,  the  relative 
wear  —  cv,  in  which  the  wear  increases  directly  with  the 
Telocity,  but  not,  however,  directly  with  the  horse-power 
transmitted. 

If  we  assume  the  coefficient  to  be  such  that  the  wear  on 
a  rope  at  1000  feet  per  minute  is  equal  to  unity,  then  the 
wear  on  the  rope  at  any  other  speed  will  be 

T.  ,  ,  .  required  speed 

Eelatlvewear  =  -    -  -. 


To  determine  the  relative  wear  per  horse-power  trans- 
mitted by  a  given  rope  at  varying  speeds,  it  will  be  neces- 


124 

sary  to  determine  the  wear  per  horse-power  for  the  rope 
running  at  any  required  speed,  and  then  divide  this  valuo 
by  the  wear  per  horse-power  when  running  at  1000  feet  per 
minute.  Let  it  be  required  to  determine  the  wear  of  a  rope 
transmitting  a  given  horse-power  at  5500  feet  per  minute, 
as  compared  to  what  it  would  be  at  1000  feet  per  minute. 

The  horse-power  transmitted  at  1000  feet  per  minute  is 
found  to  be  3.57  for  a  one-inch  rope,  at  which  speed  we 
have  assumed  that  the  wear  is  equal  to  unity,  hence  the 
wear  per  horse-power  may  be  considered  equal  to 


£57  -  -' 

At  a  speed  of  5500  feet  per  minute  the  horse-power  trans- 
mitted by  the  same  rope  is  13.53,  but  the  wear  at  this 
speed  is  assumed  to  be  five  and  a  half  times  greater  than 
at  1000  feet,  other  conditions  being  the  same,  therefore 

5  5 
the   wear    per   horse-power  =  •=-$-£$  =  .406;   that   is,   the 

lo.Oo 

wear  of  any  rope  transmitting  one  horse-power  at  5500  feet 

per  minute  is  -    —  =  1.45  times  the  wear  which  would 
.28 

occur  at  1000  feet  per  minute. 

From  this  it  will  be  seen  that  although  the  first  cost  of 
a  rope  is  cheaper  at  the  higher  speeds,  the  rope  lasts  longer 
while  running  at  the  lower  speeds  —  conditions  remaining 
constant. 

Taking  the  case  we  have  been  considering,  it  is  found 
that  the  relative  first  cost  of  the  rope  is  inversely  as  the 
horse-power  transmitted,  or 

Cost  of  rope  at  1000  feet  _  13.53  _ 
Cost  of  rope  at  5500  feet  ~  3.57 

that  is,  the  first  cost  is  3.78  times  greater  for  the  slower 
speed.  But  it  is  shown  above  that  the  rope  will  wear  out 


EOPE-DRIVItfG. 


125 


nearly  50  per  cent  faster  at  the  increased  speed;  therefore, 
taking  the  life  of  the  rope  as  well  as  the  first  cost  into  con- 
sideration, the  relative  cost  to  transmit  a  given  horse-power 

q  ryo 

at  the  speeds  noted  will  be    '      =2.6  times  greater  for  the 

1.45 

lower  speed.  This  can  be  determined  more  conveniently 
by  assuming  H  =  horse-power  transmitted  at  V  feet  per 
minute,  Hl  =  horse-power  transmitted  at  Vl  feet  per  min- 
ute, where  Ht  >  H  and  V.  >  V.  The  wear  per  horse- 

V          V 

power  in  each  case  will  then  be  proportional  to  —  and  — ; 

H  rt j 

the  relative  first  cost  of  the  rope  per  horse-power  running 
at  the  lesser  speed,  compared  to  that  when  running  at  the 
greater,  will  be  as  Hl  is  to  H\  hence  the  ratio  of  relative 
cost,  taking  wear  into  account,  is 


V 


H 


(ii) 


TABLE  XIII.— RELATIVE  WEAR  AND  COST  OF  ROPE  PER  HORSE- 
POWER TRANSMITTED. 


Velocity 
in  Feet 
per  Minute. 

Relative  Wear 
per  Horse-power 
transmitted. 

Relative  Cost  of  Rope 
per  H.  P.  transmitted, 
considering  Wear. 

1,000 

1. 

1. 

2,000 

1.03 

.54 

2,500 

1.06 

.45 

3,000 

1.10 

.40 

3,500 

1.13 

.36 

4,000 

.18 

.347 

4,500 

.25 

.345 

5,000 

.34 

.36 

5,500 

.45 

.38 

6,000 

.64 

.44 

6,500 

.93 

.52 

7,000 

2.40 

.80 

7,500 

3.22 

1.36 

126 


ROPE-DRIVING. 


Table  XIII  has  been  calculated  upon  the  above  basis  of 
comparison,  namely,  that  the  wear  in  a  rope  at  1000  feet 
per  minute  is  equal  to  unity  and  increases  directly  as  the 
speed;  and  also,  that  the  cost  of  ropes  for  a  permanent 


_ »    3.  a 


VFI  OCITY  IN  FEET  PER  MINUTE  £lt*.  World 

FIG.  45. — RELATIVE  WEAR  AND  COST  OF  ROPE  PER  H.  P. 

installation  is  proportional  to  the  square  of  the  ratio  of  the 
power  transmitted  at  different  speeds  multiplied  by  the  in- 
verse ratio  of  the  corresponding  speeds.  To  ascertain  the 
relative  wear  per  horse-power  of  a  rope  running  at  any 
given  speeds,  it  will  only  be  necessary  to  form  a  ratio 


(UNIVERSITY) 

^x^CAUFfWNUj^^ 

ROPE-DRIYINQ.  127 

between  the  values  in  the  table  corresponding  to  the  given 
speeds.  Thus  the  wear  per  horse-power  at  5000  feet  per 

1  34 

minute,  as  compared  to  that  at  2500  feet,  will  be  -^—  =  1.26. 

In  the  same  way  the  relative  ultimate  cost  per  horse- 

nn 

power  of  a  rope  running  at  these  speeds  will  be  '—  =  0.8, 

or  20  per  cent  less  for  the  greater  speed.  The  accompany - 
h]g  diagram,  Fig.  45,  represents  these  relative  values 
graphically,  to  which  is  added  the  curve  of  relative  first 
cost.  It  will  be  noticed  that  although  the  first  cost  of  a 
rope  is  a  minimum  for  a  speed  of  5500  feet  per  minute, 
when  wear  is  considered  the  minimum  cost  occurs  at  a 
speed  of  4500  feet  per  minute. 


128  ROPE-DRIVINQ. 


CHAPTER  IX. 

As  previously  noted,  it  is  desirable  in  all  cases  of  rope 
transmission  to  so  arrange  the  drive  that  the  slack  side  of 
the  rope  shall  be  on  the  upper  part  of  the  pulley,  thus  in- 
creasing the  arc  of  contact,  as  the  two  sides  will  then  ap- 
proach each  other  when  in  motion. 

In  order  that  the  desired  tensions  T^  and  Tt  shall  be 
attained  in  the  two  parts  of  a  rope,  the  deflections  or  sag 
must  be  of  predetermined  values.  The  centre  line  of  the 
rope  will  lie  in  a  curve,  which  may  be  determined  with  no 
appreciable  error  by  assuming  the  rope  to  have  no  elastic- 
ity and  to  be  of  constant  cross-section,  under  which  condi- 
tion the  curve  will  be  that  known  as  the  catenary,  the  trans- 
cendental equation  of  which  is 


y  =  o 


Let  the  form  of  curve  in  which  the  rope  hangs  be  repre- 
sented by  PO'P'  (Fig.  46),  in  which  0'  is  the  lowest  point 
of  the  rope.  Take  0'  as  the  origin  of  coordinates,  and  let 
x  =  0' A'  and  y'  =  A'P  be  the  abscissa  and  ordinate  of 
any  point  P  in  the  curve,  and  let  I  be  the  distance  between 
the  points  of  support. 

Since  we  have  assumed  the  rope  to  be  perfectly  inelastic, 
the  tension  at  any  point  of  the  curve  must  be  in  the  direc- 
tion of  the  rope.  Let  T  be  the  stress  in  the  rope  at  P,  the 
vertical  and  horizontal  components  of  which  are  repre- 
sented by  V  and  H  respectively.  Assume  the  length  of 
curve  O'P  =  s.  Since  the  weight  of  a  unit  length  of  rope 
=  w,  the  vertical  component  of  the  tension  T  is  evidently 


ROPE-DRIVING. 


129 


equal  to  one  half  the  weight  of  the  rope  between  the  points 
P  and  P',  or  V  =  iw  PO'P'  —  MS.  To  determine  the 
length  of  curve,  produce  the  tangent  through  the  point  P 
and  rectify  the  curve  O'P  on  this  tangent,  making  s  =  PQ. 


FIG.  46. 


Erect  a  perpendicular  to  PQ  at  the  point  Q,  meeting  the 
vertical  from  P  at  A ;  then  will  0.1  drawn  through  A  par- 
allel to  O'A'  be  the  directrix  to  the  catenary,  and  QA  will 
equal  A  A'  =  c.*  Hence 


-  QA\ 


Therefore 


PQ*  = 

f/)2-'2;     

V=ws  =  w  V  (ij'Y  H-  Icy'. 


=  V  (y'Y  +  icy'. 


*  For  geometry  of  this  curve  see  '  '  The  Funicular  Polygon  w  in 
Bowser's  "Analytic  Mechanics,"  p.  216  et  seq.;  also  Price's  "Me- 
chanics," vol.  i. 


130  ROPE-DRIVING. 

At  the  vertex  0  the  tension  is  horizontal  and  equal  to  the 
weight  of  a  length,  c,  of  the  rope;  but  this  tension  is  the 
same  as  the  horizontal  component  at  the  point  P;  hence 
H  =  we.  Since  T=  1/Fa  -j-  H*9  we  obtain  the  following 
value  for  the  tension  in  the  rope : 


,    .     .     (13) 

In  order  to  determine  the  parameter,  c,  let  the  equation 
e-f  the  curve  y  =  -  \ec  -|-  6~  <y  be  developed  into  the  following 
series  : 


Since  the  character  of  the  curve  is  such  that  the  quotient 

/>* 

—  is  a  proper  fraction,  the  series  will  be  converging.     Stop- 
c 

ping  at  the  third  member  as  giving  sufficient  accuracy,  we 
have 


therefore  x"  —  2  c  (y  —  c)  .......    (14) 

Substituting  the  -value  of  y  =  y'  -\-c  in  this  equation,  we 
obtain  x*  =  %cy',  which  is  the  equation  of  a  parabola  re- 

ferred to  its  axis  and  the  tangent  to  its  vertex.     Now  let  - 

« 

(Fig.  47)  equal   the  half  distance   between   supports  =  x> 
and  h  =  sag  of   the  rope  =  y'  ;   then  from    the  previous 


/ 1  Y 

equation  we  obtain  I— j  =  2ck: 

J 

8k' 


F 

hence  the  parameter  c  =  -        Substituting  this  value  of 


KOPE-DKIVItfG. 


131 


c  in  equation  (13),  we  have 

T=  «(«+*)• 

in  which  w  equals  the  weight  of  a  unit  length  of  rope  = 
0.32f?2  for  manilla ;  that  is,  the  tension  at  any  point  in  the 
rope  is  equal  to  the  weight  of  a  portion  equivalent  in  length 
to  the  parameter  plus  the  ordinate  yf  of  the  point.  From 
this  equation  we  may  obtain  the  sag  of  the  driving  or  driven 
portions  of  the  rope  by  substituting  for  T  the  values  of  Tl 
and  T^,  the  deflections  corresponding  to  which  may  be  rep- 
resented by  hl  and  h^  (Fig.  47).  T,  will  evidently  be  con- 


DRIVEN 


FIG.  47. 

stant  and  equal  to  200d2  if  our  preconceived  conditions 
are  maintained,  but  the  value  of  T^  will  be  variable,  in- 
creasing with  the  speed. 

Practically,  it  will  be  impossible  to  maintain  a  constant 
tension  in  the  rope,  so  that  the  amount  of  sag  obtained  by 
calculation  is  liable  to  vary  with  the  conditions  of  service. 
The  tension  may,  however,  be  approximately  determined 
by  the  deflection.  Assuming  the  distance  between  support- 
ing points  of  the  rope  equal  to  the  distance  between  centres 
of  pulleys,  and  solving  for  li,  we  obtain 


h  -IT*  +  1A/~T*     V 
3  ~~  2  w       %  *     ""•  —  o~ ' 


132  KOPE-DKIVING. 

The  positive  and  negative  signs  before  the  radical  in  these 
equations  indicate  two  values  for  h,  the  lesser  of  which 
only  is  to  be  used.  As  pointed  out  by  Reuleaux,  between 

1  T 

the  two  lies  a  value  h  =  -  -,  which  is  obtained  when  the 

2  w 

quantity  under  the  radical  =  0;  that  is,  when  T—— — . 

V  2 

This  deflection  is  interesting,  as  it  denotes  the  minimum 
stress  which  may  exist  in  the  rope. 

Since  the  sum  of  the  tensions  increases  with  the  speed, 
the  sag  of  the  rope  when  at  rest  is  not  directly  obtainable 
from  the  previous  values  of  hl  and  7*a. 

The  tension  necessary  for  adhesion,  which  constitutes  a 
part  of  the  stress,  T7,,  is  dependent  upon  the  speed  at 
which  the  rope  is  intended  to  be  run,  so  that  in  order  to 
determine  the  sag  7/-0  when  at  rest,  for  a  given  maximum 
tension  TJ9  the  initial  tension  must  be  obtained  for  any 
given  speed,  and  this  value  substituted  in  the  general  for- 
mula 

h  =  --  ±  -  \  — j  —  — . 

Assuming  the  tension  on  the  tight  side  of  the  rope  to 
be  made  up  of  three  parts,  namely,  the  driving  force  P, 
the  centrifugal  force  Fn9  and  the  tension  T3,  necessary  to 
balance  the  strain  for  adhesion,  we  obtain 


In  like  manner  the  tension  on  the  slack  side  of  the  rope 
may  be  assumed  to  be  produced  by  the  strain  necessary  for 
adhesion,  plus  the  strain  due  to  centrifugal  force;  that  is, 

T  —  T  4-  F 

*  i  —  •*  3    i    •*  fl- 
it is  evident  that  if  the  normal  tension  Tl  be  diminished 
by  the  centrifugal  force  the  remaining  stress  will  be  equal 
to  the  tension  necessary  for  adhesion   plus  the  driving 


ROPE- 1)  RIVING. 


133 


force;  hence  if  we  obtain  the  value  of  the  ratio  ~~  from 

e<K,  in  which  the  stress  due  to  centrifugal  force  is  neg- 
lected, we  shall  have  in  T2  the  initial  tension  necessary  for 
adhesion,  or  7'2  —  T3.  If  rl\  is  assumed  to  be  constant,  then 
7 '3  will  be  constant  for  a  given  coefficient  of  friction  and 


arc  of  contact;  therefore 


T        j__     /TT 

-—  — -3  will  equal  the  tension  in 
& 

the  rope  when  the  latter  is  at  rest. 

The  following  simpler  method  for  obtaining  the  sag, 
though  less  exact,  is  sufficiently  accurate  for  any  practical 
case  that  may  arise,  for  it  must  be  borne  in  mind  that  any 
theoretical  calculation  for  the  deflection  of  a  running  rope 
can  at  best  be  only  an  approximation,  as  it  is  exact  only 
when  the  rope  is  running  at  its  normal  speed,  transmitting 
its  full  load  and  strained  to  its  normal  tension.  Let  /  be 
the  distance  between  two  shafts  which  are  at  the  same 


FIG.  48. 


level  (Fig.  48),  and  let  li  be  the  deflection  of  the  rope;  also, 
let  CPD  =  0  be  the  inclination  of  the  rope  at  P,  from 
which 

CD  2k  4h 


sin      = 


and  hence,  since  Fis  the  vertical  component  of  the  tension 


134 


ROPE-DRIVIXG. 


w 

Tin  .the  rope  and  equal  to  — ,  where  Wi&  the  weight  of 
rope  between  supports,  we  shall  have 

~F~~T7 


sin  0 


As  h  is  small  compared  to  I,  we  may  neglect  --W,  as  it 

will  have  no  appreciable  influence  on  the  result;  then  if 
we  assume  the  length  of  rope  to  be  equal  to  the  distance 
between  the  shafts,  we  shall  have,  approximately, 

Wl_wV_ 
=  87*  ~=  87*' 

where  wl  —  W\  hence  the  sag  of  the  rope  at  the  centre 
will  be 


*=!!>• LI*] 

in  which  w  =  0.33d2  for  manilla  ropes. 

From  this  formula  the  following  table  (XIV)  has  been 

TABLE  XIV. — DEFLECTION  OF  ROPE. 

Calculated  from  h  —  ^r 

8.Z 


Deflection  in  Feet  in  Slack  Side  of  Rope 
when  r3  =  Tl  —  P  = 

Deflection 

Deflection 
in  Both 

Distance 
between 
Pulleys 
in  Feet. 

Corres 

91da 

>ponding  to 
per  Mir 

lOOd* 

Velocity  ir 
ute  of: 

112d2 
Feet 

Tight  Side 
of  Rope 
when 

Pounds. 

Sides  of 
the  Rope 
when 
Initial 
Tension 

2000 

3000 

4000 

5000 

o- 

30 

0.42 

0.39 

0.36 

0.32 

0.18 

0.25 

40 

0.74 

0.70 

0.64 

0.57 

0.32 

0.45 

60 

1.67 

1.58 

1.44 

1.28 

0.72 

1.02 

80 

2.97 

2.81 

2.56 

2.28 

1.28 

1.82 

100 

4.65 

4.40 

4.00 

3.57 

2.00 

2.84 

120 

6.70 

6.33 

5.76 

5.14 

2.88 

4.10 

140 

9.12 

8.61 

7.84 

7.00 

3.82 

5.58 

160 

11.90 

11.25 

10.24 

9.14 

5.12 

7.27 

ROPE-DRIVING. 


135 


computed  and  the  curves  plotted,  as  shown  in  Fig.  49,  Tl 
being  assumed  as  constant  and  equal  to  200^2  pounds. 
The  tension  T^  which  varies  with  the  speed,  has  been 
separately  determined  for  the  several  cases  considered  from 

T 

-L  =  eWi-«), 

•*  2 

in  which  the  coefficient  of  friction,  0,  =0.31  and  a  =  2.88; 
hence 

12  Ft. 


40        60        80       100       120       140     160  Ft. 

DISTANCE  BETWEEN  CENTERS  OF  PULLEYS.  Elec.  World 

FIG.  49. — DEFLECTION  OF  ROPES. 


136  ROPE-DRIVING. 

log  ^  =  0.4343  X  0.9(1  -  z). 
•f  « 

The  deflection  on  the  tight  side  will  then  be 


'  -       ST      ~  8X00 
and  on  the  slack  side  &2=  -?p—>  in  which  T7,  has  the  values 

-^  2 

given  in  the  table. 

As  previously  pointed  out,  the  initial  tension  in  the  rope 
when  at  rest  may  be  obtained  by  neglecting  F^  ,  in  which 

T 

case  we  have  log   ~  =  0.  43430**,  from  which  we   find 

*  2 

^  =  2.46,  or  T;  =  82d2;  and  since  T3  =  Tt  when  there  is 

•*  * 

no   centrifugal   force   acting,  the   initial  tension  T%  will 

equal 


hence  T9  =  141^2.  From  this  we  may  obtain  the  deflec- 
tion when  the  rope  is  at  rest,  as  noted  in  the  last  column 
of  Table  XIV,  which  has  been  computed  from 

ft  = 


where  I  is  the  distance  in  feet  between  centres. 

To  draw  the  curve  of  the  rope  in  order  to  determine 
the  space  it  will  occupy,  assume  it  to  hang  in  a  parabola 
with  origin  at  vertex  and  lay  off  the  X  and  Y  axes,  as  in 
Fig.  50.  Let  05  =  %l  =  -J-  distance  between  points  of  sus- 
pension of  the  rope,  and  let  5H  =  h,  the  sag  of  rope.  Di- 
vide 5H  into  a  convenient  number  of  equal  parts,  also 
divide  05  into  the  same  number  of  equal  parts;  erect  per- 
pendiculars from  the  points  of  division  1,  2,  3  ...  and  join 


ROPE-PRTVIKft. 


13? 


1',  2',  3'.  .  .  with  the  origin  0.  The  curve  drawn  through 
the  points  of  intersection  will  represent  one  branch  of  the 
parabola  desired. 

In  outdoor  drives,  where  the  configuration  of  the  ground 
would  prevent  the  proper  amount  of  sag  being  used,  or,  in 


o 
FIG. 


1  23 

50. — METHOD  OF  LAYING  OUT  CURVE. 


4  5  , 

Eke.  World 


TV     iV 


FIG.  51. — INCLINED  TRANSMISSION, 

general,  where  obstructions  prohibit  the  proper  deflection 
for  a  given  distance  between  centres  of  shafts,  the  rope 
should  be  carried  on  supporting  pulleys  or  sag-carriers, 
examples  of  which  have  already  been  given. 


138  HOPE-DRIVING. 

Where  the  pulleys  are  placed  at"  different  heights  we 
have  an  inclined  transmission,  and  the  curve  in  such  cases 
is  unsymmetrical,  as  in  Fig.  51.* 

This  curve  is  best  solved  by  an  approximation  similar  to 
that  already  given. 

It  will  be  noticed  that  the  curve  of  the  rope  ABC  is 
made  up  of  two  unequal  parts,  AB  and  BC,  whose  hori- 
zontal projections  are  I'  and  I"  and  their  corresponding 
deflections  h'  and  h".  The  given  difference  in  height  be- 
tween the  points  of  support  of  the  rope  is  ED  =  h  —  lif  — 
li"  ,  and  their  known  horizontal  distance  is  1=1'-}-  I"  t 
From  the  preceding  formula  for  the  deflection  (equation 
[15]  )  we  have 

/»"    and    A"  =    ?      =  /*»-, 


in  which  fi  is  taken  as  a  coefficient  depending  upon  the 

value  of  the  tension  in  the  rope  —  -5-^. 

o  J. 

Since  h  =  /*'  -  li"  and  I  =  V  +  1",  we  have 


therefore  I'  -  I"  =  ~.     But  V  =  I  -  I"  and  I"  =  I  -  I'-, 
hence   2/311'  —  01*  =  h,  and   the   required   distance  I'  = 

Toy    .    In  the  same  way  V9  =  —  -^—^  —  ,  and  the  deflections 
lipl  Zpl 

are 


in  each  of  which  equations  fi  has  a  separate  value  depend- 
ing upon  the  tension  in  the  rope.  It  is  evident  that  the 
tension  T'  at  A  will  be  greater  than  that  at  G,  on  account 
of  the  greater  weight  of  rope  lying  in  the  upper  branch  of 

*  Weisbach,  vol.  m.  p.  243. 


ROPE-DRIVING. 


130 


the  parabola.  If  T'  is  known,  the  lesser  tension  T"  am 
be  determined  by  assuming  the  length  of  the, rope  to  be 
the  same  as  if  the  points  A  and  C,  Fig.  52,  were  at  the 
same  height  and  I  feet  apart.  From  previous  considera- 


FIG.  52. 

tion  it  is  seen  that  the  tension  at  any  part  of  the  rope  in 
a  parabola  is  equal  to  the  weight  of  rope  equivalent  in 
length  to  the  parameter  c  added  to  the  ordinate  of  the 
point;  hence  if  h"  equals  the  ordinate  of  Q  —  li'  —  h,  we 
shall  have 

T"  =  w(c  +  h")     and     T'  =  w(c  +  //). 


Since  c=, 


and 


therefore  T"  -  wli"  =  T'  -  wli'  .     Substituting  for  h'  its 
value,  h  -\-  Ji",  there  is  obtained 

T"  =  T  -  wh. 

When  a  tension-carriage   is   used  the  necessary  weight 
can   be   ascertained  from  the  formula  for  back   tension, 


140  ROPE-DRIVIHG. 

?\  =  Tt-\-F9;   if  we  assume  F0  =  0,  we   shall  have    the 
initial  tension  necessary  for  adhesion  equal   to  the   ten- 

T 

sion  in  slack  side  of  rope,  that  is,  T3  =  Tt.     But  Ty  =  — L- 

/w.4o 

T 

(from   log   -=£  =  0.4340O',  assuming  previous  values  of  0 

•*•  2 

and  a);  therefore  the  initial  tension  in  the  rope  will  be 

T,   _  200flf 
3  ~  2.46  ~  2.46  ' 
When  diameter  of  rope  equals 

f  inch  the  initial  tension  =    31  pounds. 

3        «  {(  «  ((  __       Aft  « 

£     "      «       "  «        —    62       " 

^  «          ((  <(  a  __      go          (( 

11        «          f(  ((  ((  __    19I7          *f 

1£     "       "       "  "       =  184       " 

If     "       "        "  «        =  250       " 

2       a       '(        C(  (l        ^  3^5       4< 

The  actual  weight  to  be  placed  on  the  tension-pulley  will 
depend  upon  the  arrangement  of  drive,  and  in  order  to 
obtain  the  best  results  should  be  determined  for  each  par- 
ticular case.  It  is  to  be  noted  that  with  a  vertical  tension- 
carriage  the  weight  of  carriage  and  sheave  must  be  consid- 
ered as  a  portion  of  the  weight  producing 
3  tension  in  the  rope. 

If  W  equals  the  total  weight  necessary 
to  maintain  the  tension  Tt  in  each  part 
of  the  rope  leading  off  from  the  tension- 
pulley,  then,  when  these  two  portions  are 
parallel,  we  shall  have  W=2TZ-,   but  if 
the  rope  leads  from  the  tension-pulley  so 
that  it  includes  an  angle  2#  between  its 
sides,  as  shown  in  Fig.  53,  then  the  weight  W  will  be  less, 
and  may  be  found  from  W=  2jP3  cos  0. 


ROPE-DRIVING. 


CHAPTER  X. 

IT  has  been  previously  stated  (see  page  7)  that  where 
rope-driving  is  used  the  loss  at  the  engine  in  friction  may 
be  taken  in  a  general  way  as  about  10  per  cent  of  the  rated 
horse-power  of  the  engine,  that  an  additional  10  per  cent 
is  absorbed  by  the  shafting,  and  that  from  5  to  8  per  cent 
may  be  attributed  to  losses  in  the  rope  itself,  due  to  resist- 
ance to  bending,  wedging  in  the  grooves,  differential  driv- 
ing effect,  and  creep,  all  of  which  affect  the  loss  to  a  greater 
or  less  extent. 

According  to  the  accepted  laws  of  solid  friction  we  should 
expect  that  with  an  increased  load  on  the  engine  the  fric- 
tion would  be  increased  in  direct  proportion  to  the  load, 
but  in  nearly  all  experiments  to  determine  the  friction  of 
steam-engines  we  have  the  anomaly  that  the  work  neces- 
sary to  overcome  friction  is  practically  constant,  and,  in 
fact,  in  many  cases  it  is  a  little  greater  when  running  with- 
out load  than  when  the  engine  is  fully  loaded. 

However,  it  is  probable  that  the  ordinary  laws  of  friction 
obtain  here  as  in  other  cases  of  sliding  and  rolling  con- 
tact; but  instead  ol  having  a  constant  coefficient  of  friction 
for  the  surfaces  in  contact,  the  coefficient  may  be  con- 
sidered as  a  variable  depending  upon  the  degree  and  distri- 
bution of  lubrication.  The  lubrication  of  engine  bearings 
(both  rolling  and  sliding  contact),  approaches  more  nearly 
to  the  condition  which  obtains  when  the  bearings  are  sub- 
jected to  an  oil  bath,  and  although  the  lubrication  is 
restricted,  yet  the  result  is  similar  in  action. 

Experiments*  on  the  friction  of  a  well-lubricated  journal 

*Proc.  I.  M.  E.  November,  1885;  also  Kent's  MecUauiculEugiiieer's 
Pocket  Book, 


142  ROPE-DRIVING. 

(oil  bath),  show  that  the  absolute  friction,  that  is,  the  ab- 
solute tangential  force  per  square  inch  of  bearing,  required 
to  resist  the  tendency  of  the  brass  to  go  round  with  the 
journal,  is  nearly  a  constant  under  all  loads  within  ordinary 
working  limits.  Most  certainly  it  does  not  increase  in 
direct  proportion  to  the  load,  as  it  should  do  according  to 
the  ordinary  theory  of  solid  friction.  The  results  of  these 
experiments  seem  to  show  that  the  friction  of  a  perfectly 
lubricated  journal  follows  the  laws  of  liquid  friction  much 
more  closely  than  those  of  solid  friction.  They  show  that 
under  these  circumstances  the  friction  is  nearly  indepen- 
dent of  the  pressure  per  square  inch,  and  that  it  increases 
with  the  velocity,  though  at  a  rate  not  nearly  so  rapid  as 
the  square  of  the  velocity. 

The  experiments  on  friction  at  different  temperatures 
indicate  a  great  diminution  in  the  friction  as  the  tempera- 
ture rises.  Thus  in  the  case  of  lard-oil,  taking  a  speed  of 
450  revolutions  per  minute,  the  coefficient  of  friction  at  a 
temperature  of  120°  is  only  one-third  of  what  it  was  at  a 
temperature  of  60°.  In  regard  to  engine  friction,  whatever 
be  the  cause,  it  is  a  well-known  fact  that  the  coefficient  of 
friction  decreases  as  the  load  increases,  so  that  at  all  ordi- 
nary speeds  the  internal  resistance  of  the  engine  may  be  con- 
sidered sensibly  constant,  in  which  case  the  so-called  fric- 
tion-card of  the  engine  represents  practically  the  friction 
of  the  machine  when  fully  loaded — the  indicated  power 
without  load  being  sensibly  the  measure  of  the  wasted  work 
of  the  engine  when  in  operation  under  load  of  whatever 
amount. 

That  is,  the  engine  friction  is  independent  of  the  load 
and  is  a  function  of  the  characteristic  of  the  engine  itself, 
of  the  speed  of  piston  and  rotation,  and,  to  a  slight  extent, 
of  the  steam-pressure  and  of  the  method  of  steam-distri- 
bution :  so  that  while  we  may  speak  of  the  friction  as  being 
a  certain  percentage  of  the  horse-power  of  an  engine,  it 


ROPE- DRIVING. 


143 


must  be  understood  to  refer  to  the  rated  indicated  horse- 
power ;*  at  less  than  rated  power  the  percentage  of  loss  due 
to  friction  will  be  greater,  and  at  maximum  power  the  per- 
centage will  be  less.  This  is  shown  by  way  of  illustration 
in  the  following  table  (XV),  which  is  taken  from  a  paper 
presented  by  Prof.  Thurston  before  the  American  Society 
of  Mechanical  Engineers  in  1886  :f 

This  engine,  a  "  Straight  Line,"  was  8  inches  in  diameter 
of  cylinder  by  14  inches  stroke;  it  had  a  balanced  valve 
with  stroke  varying  from  2  to  4  inches  according  to  position 
of  governor  and  eccentric,  a  fly-wheel  50  inches  in  diam- 
eter, weighing  2300  pounds.  Its  rated  load  was  35  to  40 
horse-power. 

TABLE  XV.— FRICTION  PER  CENT  UNDER  VARYING  LOADS. 


Revolutions 
per 
Minute. 

Steam 
Pressure. 

Indicator 
Horse-power. 

Friction 
Horse-power. 

Friction 
per  cent. 

232 

50 

7.41 

3.35 

45 

229 

65 

7.58 

2.60 

34 

230 

63 

10.00 

4.00 

40 

230 

73 

11.75 

3.65 

32 

230 

75 

14.02 

4.02 

28 

230 

80 

15.17 

3.17 

21 

230 

75 

16.86 

2.86 

17 

230 

75 

28.31 

3.36 

11.75 

229 

60 

33.04 

3.16 

9.5 

229 

58 

37.20 

2.34 

6.3 

229 

70 

43.04 

3.19 

7.4 

230 

85 

47.79 

2.75 

5.8 

230 

90 

52.60 

2.60 

4.9 

230 

85 

57.54 

2.54 

4.4 

Examining  the  above  table,  it  will  be  seen  that  the  fric- 
tion of  the  engine  varies  somewhat  with  varying  steam- 
pressures  and  total  power,  but  in  such  a  manner  as  to  indi- 
cate the  controlling  cause,  as,  for  instance,  imperfect  lubri- 

*  Thurston,  Trans.  A.  S.  M.  E.,  vol.  x.  p.  110. 
f  Trans.,  vol.  vm.  p.  0°, 


144  KOPE-DBIVING. 

cation,  to  be  irregular  iu  action,  and,  possibly,  to  some 
extent,  due  to  errors  of  observation  and  to  accident.  The 
average  friction  horse-power  is  3.11  h.  p.,  and  the  variations 
from  this  value  are  distributed  throughout  the  whole  series, 
showing  that  the  work  necessary  to  overcome  the  friction 
is  practically  constant,  and  independent  of  the  load.  The 
friction  of  this  engine  is  quite  low,  as  at  its  normal  rating 
the  percentage  of  loss  is  less  than  7  per  cent.  According 
to  D.  K.  Clark,*  the  frictional  resistance  of  steam-engines 
varies  from  8  to  20  per  cent  of  their  normal  indicator  h.  p. 
—the  size  of  engines  experimented  upon  ranging  from  13 
to  350  h.  p. 

In  recent  tests  of  American  engines  it  has  been  shown 
that-with  first-class  workmanship  and  balanced  valves  the 
percentage  of  loss  at  normal  working  load  may  be  reduced 
to  about  6  per  cent,  both  with  Corliss  and  with  high-speed 
single-cylinder  automatic  engines;  but  this  is  exceptional, 
and  we  may  expect  with  ordinary  lubrication  that  the  fric- 
tional resistance  will  vary  from  8  to  13  per  cent  of  the 
normal  load. 

Compound  engines  of  the  better  class  should  not  absorb 
more  than  about  10  per  cent,  and  triple-expansion  engines 
no  more  than  12^  per  cent,  under  full  load. 

For  small  engines,  either  single  or  compound,  there  is 
probably  little  difference  between  the  internal  resistance, 
whether  geared  with  ropes  or  flat  leather  belts,  for  the 
weights  of  fly-wheel  and  grooved  pulley  and  the  diameter 
of  shaft  would  be  essentially  the  same  in  each  case.  With 
larger  engines,  however,  the  belts  would  require  a  wider- 
faced  fly-wheel,  which,  on  account  of  the  greater  distance 
between  bearings,  would  necessitate  a  larger  shaft,  and 
hence  increased  work  in  overcoming  journal-friction — as- 
suming the  same  weight  of  wheels  and  speed  of  rotation. 

*  "  The  Stecam  Eugiiie,"  vol.  n.  p.  616, 


ROPE-DRIVING. 


145 


With  the  rope-driving,  also,  the  speed  of  the  rim  will  be 
greater,  and  a  somewhat  lighter  pulley  may  be  used  to 
insure  the  same  degree  of  steadiness  in  running;  in  many 
cases  the  ropes  are  delivered  from  the  fly-wheel  in  a  nearly 
vertical  direction,  so  that  a  certain  portion  of  the  weight 
on  the  hearings  is  neutralized  by  the  upward  pull  of  the 
ropes.  Moreover,  the  elasticity  and  recoil  of  the  ropes  act 

n 


FIG.  54. 

in  the  same  manner  as  mass  in  the  rim,  and  for  this  reason 
a  lighter  wheel  may  be  used  with  rope-driving  provided 
the  construction  is  such  as  will  permit  it.  The  journal- 
friction  is  therefore  presumably  less  in  the  larger  class 
of  engines  employing  ropes,  when  compared  with  those 
using  belts.  The  difference  is,  however,  not  great ;  and 
since  the  actual  resistance  in  any  case  is  also  dependent 
upon  peculiarities  of  type  and  construction,  the  lower 
values  of  engine-friction  previously  given,  viz.,  8  to  10  per 
cent  of  normal  horse-power,  may 

S    T^-       OF  THE 

TQTNIVERSITT 


146 


ROPE-DRIVING. 


good  for  rope-driving  plants  of  medium  size,  while  9  to  12 
per  cent  of  the  normal  horse-power  may  be  taken  as  suit- 
able for  the  larger  class  of  engines  running  under  favor- 
able conditions. 

In  the  ordinary  transmission  of  power  by  shafting  we 
find  the  shaft  loaded  with  pulleys  and  the  power  taken  off 
in  varying  amounts  throughout  its  entire  length;  it  is  un- 
usual except  in  short  lengths  to  receive  the  power  at  one 
end  and  transmit  it  at  the  other.  Moreover,  in  long  shaf  t- 


FIG.  55 

ing  the  head  or  receiving  shaft  is  usually  situated  midway 
between  the  ends,  and  the  power  distributed  more  or  less 
uniformly  from  this  head  shaft  to  either  end ;  therefore, 
in  estimating  the  power  absorbed  by  friction  in  ordinary 
mill  or  factory  shafting  loaded  with  pulleys  the  previous 
formulas  (page  70)  do  not  apply,  as  these  relate  only  to 
those  cases  where  power  is  taken  off  at  the  end  of  the  shaft. 
The  conditions  of  practice  as  we  find  them  in  actual 
transmissions  are  so  various,  that  it  is  difficult  to  lay  down 
any  general  rule  by  which  the  power  absorbed  by  friction 


ROPE-DRIVING.  147 

may  be  determined:  the  number  and  weight  of  pulleys 
and  couplings,  the  intensity  and  direction  of  belt-pull,  the 
condition  of  bearings  and  their  lubrication,  till  affect  the 
amount  of  work  lost  in  friction. 

For  the  ordinary  factory  shafting,  from  which  power  is 
taken  fairly  uniformly  throughout  its  length  and  dis- 
tributed horizontally  to  counter-  or  auxiliary  shafts  situ- 
ated on  one  or  both  sides  of  the  main  shaft,  there  will  be 
three  general  cases  to  be  considered,  as  shown  in  Figs.  54, 
55,  and  56,  and  eacli  of  these  cases  will  be  modified,  depend- 
ing upon  the  direction  of  the  belt  to  and  from  the  main 
shaft. 

For  our  present  purposes  it  will  be  sufficient  to  take  that 
case  in  which  the  shaft  friction  is  a  maximum  for  the 
assumed  direct-ion  of  main  belt-pull  corresponding  to  the 
arrangement  shown  in  Fig.  54. 

The  friction  will  evidently  be  proportional  to  the  weight 
of  the  shaft  and  the  unbalanced  belt-pull  acting  on  the 
shaft. 

The  weight  of  pulleys,  belts,  clutches,  and  couplings 
carried  by  the  line-shaft  will  vary  from  about  one  and  one- 
half  to  three  times  the  weight  of  shaft,  so  that  the  total 
weight  on  the  bearings  will  vary  from  two  and  one-half  to 
four  times  the  weight  of  shaft;  for  head  and  jack  shafts 
the  total  weight  will  probably  vary  from  three  to  five  times 
the  weight  of  shaft. 

In  addition  to  this  weight  there  is  the  unbalanced  belt- 
pull,  which  increases  the  load  on  the  bearings.  Although 
the  tension  on  the  tight  side  of  the  belt  should  not  ordi- 
narily exceed  about  twice  the  tension  in  the  slack  side 
necessary  for  adhesion,  yet  it  is  probable  that  belts  are  fre- 
quently run  with  a  ratio  of  tension  equal  to  one  to  three, 
and  occasionally  one  to  four;  on  the  other  hand,  it  is  a 
very  common  thing  for  belts,  especially  short  ones,  to  be 
laced  so  taut  that  the  initial  tension  is  greatly  in  excess  of 


148 


ROPE-DRIVING. 


that  required  for  adhesion,  iu  which  case  the  sum  of  the 

tensions  approaches  twice  that  in  the  tight  side  of  the  belt.* 

With  ordinary  shop-worn  belting  it  will  be  safe  to  assume 

that  the  tension  fl\  on  the  slack  side  of  the  belts  is  one  half 

T 

the  tension  T7,  on  the  tight  or  driving  side,  that  is,  rl\  =  — '; 

& 

hence,  since  T7,  —  T9  =  P,  the  driving  force,  we  have 

I'X  33-000- 
The  velocity  of  intermediate  belting  is  so  variable  that 


FIG.  56. 


any  assumption  of  speed  must  be  regarded  as  applying  to 
a  particular  case  or  representative  of  a  certain  type  of 
factory,  and  cannot  be  taken  as  general.  In  many  ma- 

*This  is  often  a  source  of  much  trouble,  as  the  increased  tension 
not  only  increases  the  loss  due  to  friction  but  in  many  instances  the 
useful  power  transmitted  is  not  sufficient  to  drive  the  machine.  In 
such  cases,  by  slacking  out  the  lacing  or  inserting  a  short  piece  of  belt 
so  as  to  reduce  the  tension,  heavy  cuts  can  readily  be  taken  when  it  is 
practically  impossible  to  run  the  machine  empty  with  the  tight  belt, 


140 

chine-shops  the  average  speed  of  intermediate  belts  is  not 
more  than  500  feet  per  minute  ;  in  others  the  average 
speed  is  more  than  twice  as  great,  and  in  wood-working 
shops  it  is  still  greater. 

For  our  present  purpose  we  shall  assume  an  average 
speed  of  660  feet  per  minute  for  belts  running  from  the 
main  shaft  to  a  secondary  or  counter  shaft. 

T          V 

Substituting  this  value   in  HP  =  —-  X          —  ,  there   is 

6         ooOOO 


obtained  T.  =    ---         HP  =  100  X  HP',  but  since  the 
060 

horse-power  which   the   shaft  is  capable  of  transmitting 

(FN 
may  be  considered  equal  to  —  —  ,  where  d  is  the  diameter  of 

shaft  in  inches  and  ^Vthe  number  of  revolutions  per  minute, 
we   have  the  tension  on  the  tight  side   of  all   the  belts 

d'A  Y 
^  =  100  X  ~^  =  d*Nm,  therefore  the   sum  of  tensions 

+  TJ  =  ld3N,  and  the  pull  per  foot  of  length  of 

«Q 

Zd'N 
shaft  =-.-nr. 

In  the  present  case  it  will  be  noted  by  reference  to  Fig. 
54  that  there  is  an  additional  pull  on  the  bearings  due  to 
the  tensions  in  the  belt  from  fly-wheel  to  main-line  shaft. 
If  the  ratio  of  tight  to  slack  side  tension  remains  the  same 
as  before,  and  we  consider  that  the  velocity  of  main  belt  is 
four  times  as  great  as  that  of  the  intermediate  belting,  the 
additional  belt-pull  will  equal  approximately  one  fourth  of 
the  sum  of  the  belt-pulls  from  the  main  to  the  counter 
shafts  or  machinery.  The  resultant  of  these  tensions, 
combined  with  the  weight  of  shafting  and  pulleys,  will  be 
the  effective  load  on  the  bearings. 

Assuming  an  angle  of  30°  with  the  horizontal  for  the 
line  of  action  of  the  resultant  pull  on  the  bearings  due  to 


150  ROPE-DRIVING. 

the  tensions  in  the  tight  and  slack  sides  of  the  main  belt, 
the  combined  horizontal  pull  on  the  bearings  will  be 


COS 


and  the  vertical  pull  will  be,  when  W8  =  weight  of  loadeu 
shaft, 


Therefore   the  resultant  of  both  horizontal  and  vertical 
forces  acting  on  the  bearings  will  be 


W=          w      sin  30°  X  - 


As  previously  shown,  the  horse-po"wer  necessary  to  over- 

come journal-friction  will  be  HP0  =  ,  where  F  is  the 

ooUUU 

force  of  friction  at  the  circumference  of  shaft  and  v  is  the 
speed  in  feet  per  minute  of  a  point  on  the  circumference. 
If  the  bearing  is  well  worn  and  fitted  to  its  shaft,  the  resist- 
ance due  to  friction  will  probably  lie  between  the  limits 

—  0JFand  —  0TF,  where  0  is  a  coefficient  which,  from  the 

M  7t 

results  of   experiments  on  shafting  with  ordinary  lubri- 
cation,  we   have   assumed   equal   to   0.06,  and   W  is   the 
resultant  load,  in  pounds,  on  the  bearings. 
From  the  lesser  of  these  values  there  is  obtained 


(16) 


But  we  have  assumed  that  the  weight  of  a  loaded  shaft 
varies  from  two  and  one-half  to  four  times  the  weight  of 
shaft;  taking  an  average  value  of  three  for  line-shafting, 
and  noting  that  the  weight  of  shaft  per  foot  of  length 


ROPE-DRIVING.  151 

equals  —  (3.36cf),  we  have  the  friction  on  a  loaded  shaft  L 
feet  long,  due  to  its  weight  = 

A  I  fT  \ 

-031-  X  3.36f/a]Z, 
n      \4  / 

Substituting  the  value  of  W  in  formula  (16)  when  the 
belt  tensions    are   taken   into    account,   and   noting   that 

Ws  —  3  ^  (3.36d2.£),  we  have  for  the  total  friction  load 


su 


From  the  formula  for  the  power  absorbed  by  friction  we 
obtain 


lience  the  ratio  of  power  absorbed  by  friction  to  the  horse- 
power which  the  shaft  is  capable  of  safely  transmitting 
will  be 

HP        0.058dNF       0.08F 

0612^  =     -^-Percent.  .     .    (1,) 


, 


From  this  expression  the  following  table  (XVI)  has  been 
computed  for  the  given  diameters  and  lengths  of  shafting 
running  at  100  and  250  revolutions  per  minute,  the  belt 
speed  for  the  secondary  belts  being  assumed  at  an  average 
of  G60  feet  per  minute. 

For  intermediate  belts  having  a  greater  average  velocity 
than  that  assumed,  viz.,  6GO  feet  per  minute,  the  friction 
horse-power  for  a  given  number  of  revolutions  will  be  less 
than  that  given  in  the  table.  Thus  if  the  average  velocity 
of  cross-belts  equals  2640  feet  per  minute,  the  horse-power 
transmitted  being  the  same,  it  follows  that  the  tensions  in 
the  secondarv  belts  will  be  one-  fourth  of  that  obtained  with 


152 


ROPE-DRIVING. 


the  lesser  speed;  if  the  main  driving-belt  have  the  same 
velocity,  the  tension  in  this  belt  may  be  considered  equal  to 
that  existing  in  the  intermediate  belts:  therefore,  as  the 
velocity  of  the  belt  increases  for  a  given  speed  of  rotation 
the  sum  of  the  tensions  acting  on  the  bearings  will  decrease, 
and  the  maximum  horse-power  transmissible  by  the  shaft 
will  be  exerted  with  a  decreasing  friction  loss. 

TABLE  XVI. — POWER  ABSORBED  BY  FRICTION  IN  LINE- SHAFT. 


Diameter  of 
Shaft 
in  Inches. 

Revolutions 
per 

Minute. 

Percent) 
100 

ige  of  Loss  w 
200 

hen  Length  ii 
300 

i  Feet  = 
400 

2 

2* 
3 
3* 

(100 

1250 
j  100 
|250 
100 
'  250 
100 
'  250 

5.5 

7.8 
5.8 
8.9 
6.1 
9.9 
6.8 
11.4 

10.1 
11.6 
10.3 
12.4 
10.4 
13. 
10.6 
14,2 

is" 

16.6 
15 
17.2 
15.4 

18 

ti',6 
21.5 
20 
22 

Line  of  action  of  resultant  of  main-belt  tensions  =  30°. 

Velocity  of  main  belt  2640  feet  per  minute. 

Belts  from  line  shaft  are  horizontal  and  run  at  an  average  of  660 
feet  per  minute. 

All  the  belts  are  assumed  to  drive  from  one  side  of  the  shaft  toward 
the  engine. 

Weight  on  bearings  three  times  weight  of  bare  shaft. 

If  in  any  case  the  shafts  are  belted  vertically  or  at  any 
other  angle  than  that  assumed,  the  formula  for  F  will  be 
modified  accordingly. 

For  a  head  or  jack  shaft  carrying  heavier  pulleys  the 
weight  acting  on  the  bearings  may  be  taken  equal  to  four 
times  the  bare  weight  of  shaft;  in  which  case,  other  condi- 
tions remaining  the  same,  we  obtain 

sin  30°  x  ^ 

Since,  however,  the  extra  weight  of  pulleys  on  a  jack- 
shaft  is  liable  to  produce  a  greater  bending  moment,  it  is 
customary  to  assume  a  larger  shaft  to  transmit  a  given 


KOPE-DRIVIKG.  153 


horse-power;  therefore,  instead  of  using  —  —  as  the  working 

100 

horse-power  transmissible  by  the  shaft,  it  is  better  to  use 

tTN 
under  these  conditions  HP  =  TT>~~* 

From  this  value  of  the  power  transmitted  we  obtain 

F=  n^V  [4|(3.36d2)L  +  sin  30°  x  i(|x.8dW)]V[l.s(|x.8cPJV)]8 

=   tf\Q.8d*L  -f  0.0114eZ3jy]2  +  [O.lleM]2. 

The  ratio  of  power  absorbed  by  friction  to  the  horse- 
power which  the  shaft  is  capable  of  safely  transmitting  will 
now  become 

HP       Q.tfSdNF     Q.WF 


HP 

from  which  Table  XVII  has  been  determined.  In  calcu- 
lating the  values  given  in  this  table  it  has  been  assumed 
that  the  belt  speeds  are  the  same  as  those  previously  con- 
sidered, namely,  2640  feet  per  minute  for  main  belt  to 
head  shaft,  and  one  fourth  of  this,  or  660  feet  per  minute, 
from  head  shaft  to  auxiliary  shafting.  This  may  be  low  in 
many  cases,  but,  as  already  pointed  out,  the  force  F  will 
decrease  under  the  assumed  conditions  as  the  belt  speed 
increases,  so  that  we  may  expect  the  friction  loss  obtained 
by  the  above  formula  to  be  somewhat  less  than  the  actual. 
In  the  foregoing  discussion  it  has  been  assumed  that  the 
shaft  transmits  its  allowable  maximum  power,  that  is,  for 

d*N 

line-shafting  the  power  transmitted  =  77^-,  and  for  head- 

100 

d*N 
shafts  the  power  transmitted  =  —  —  -. 

1/vO 

As  a  general  thing,  the  actual  average  power  transmitted 
by  a  shaft  is  not  more  than  about  three-fourths  of  its  as- 
sumed working  capacity;  and  since  the  weight  and  speed 
remain  practically  constant,  the  percentage  of  loss  under 
conditions  approximating  those  we  have  assumed  will  be 
somewhat  greater  than  that  given  in  the  table.  But  while 


154 


BOPE-DRIVIKG. 


the  power  transmitted  may  be  diminished  25  per  cent,  the 
percentage  of  increase  in  friction  loss  will  vary  between  wide 
limits,  depending  upon  the  speed  of  rotation  and  length  of 
shaft.  Thus  for  a  3"  head-shaft  100  feet  long,  delivering 
three-fourths  of  its  allowable  capacity  at  100  revolutions 
per  minute,  the  loss  increases  from  9.0  to  11.5  per  cent, 
which  represents  a  gain  of  28  per  cent;  at  250  revolutions 
the  loss  is  now  14.0  per  cent,  corresponding  to  an  increase 
of  14.3  per  cent;  while  at  400  revolutions  the  loss  is  17.8 
per  cent — a  gain  of  only  10.6  per  cent. 

TABLE  XVII. — POWER  ABSORBED  BY  FRICTION  IN  JACK  OR 
HEAD  SHAFTS. 


Diameter  of 
Shaft 
in  Inches. 

Revolutions 
per 
Minute. 

Percentage  of  Loss  w 
^            50 

hen  Length  in  Feet  = 
100 

(100 

4.8 

8.5 

2 

^'250 

7.2 

10.1 

(400 

10.0 

12.5 

(  100 

5.1 

8.7 

2* 

^250 

8.3 

11.0 

400 

12.2 

14.3 

(  100 

5.5 

9 

3 

^250 

9.5 

12 

(400 

14.2 

16.1 

(100 

5.8 

9.1 

8* 

^250 

11.5 

12.9 

(400 

16.6 

17.8 

(100 

6.3 

9.5 

4 

•hso 

12.1 

14.2 

400 

18.4 

20.5 

For  these  determinations  it  is  supposed  that  the  shafting 
is  properly  supported,  with  hangers  sufficiently  close  to 
each  other  to  prevent  undue  deflection  under  working  con- 
ditions, and  that  the  shaft  is  in  line,  having  good  bearings 
lubricated  as  in  common  practice.*  Departures  from  these 

*In  the  above  discussion  it  was  assumed  that  the  coefficient  of 
friction  is  constant  and  that  the  friction  varies  directly  as  the  .load. 
While  recent  experiments  on  machine  journals  running  in  oil  indi- 
cate that  the  coefficient  of  friction  varies  inversely  with  the  load 
there  seems  no  good  reason  to  doubt  the  truth  of  Morin's  laws  for 


HOPE-DRIVIKG.  155 

assumptions  will  further  increase  the  friction  loss;  but, 
on  the  other  hand,  this  loss  will  be  decreased  if  lighter  or 
fewer  pulleys  be  used  throughout  the  length  of  the  shaft, 
if  the  bearings  be  continuously  lubricated,,  or  if  the  ma- 
chines be  belted  directly  from  the  shaft  below.  "Where 
shafting  is  employed  there  will  generally  be  an  additional 
loss  due  to  the  friction  of  the  auxiliary  shafting  and  coun- 
ter-shafts, which  is  extremely  variable. 

It  is  outside  the  province  of  the  present  subject  to  discuss 
the  losses  in  these  secondary  shafts:  the  losses  which  we 
have  here  been  considering  are  those  which  exist  in  main 
line-shafting,  jack-  and  head-shafts  receiving  their  power 
presumably  by  leather-belting  or  ropes,  with  either  of  which 
for  similar  drives  there  should  be  no  appreciable  difference 
in  the  total  weights  of  pulleys  and  shafting  and  the  friction 
involved. 

The  diameter  of  grooved  shaft-pulleys  will  be  larger  and 
the  rim  will  be  thicker;  but  for  the  same  horse-power  trans- 
mitted the  width  will  be  less  and  the  weight  not  materially 
increased.  In  any  case  the  total  weight  of  grooved  pul- 
leys compared  with  that  of  belt-pulleys  used  in  the  same 
system  is  very  small,  and  any  individual  differences  may 
be  neglected  when  taken  as  a  whole.  In  those  cases  where 
ropes  are  used  exclusively,  as,  for  instance,  in  dynamo  rooms 
and  other  power-stations,  the  pulleys  are  frequently  heavier, 
and  the  shafting  usually  is  fitted  with  a  number  of  friction- 
clutches,  thus  materially  increasing  the  weight  on  the  bear- 
ings; in  cotton-mills  also,  where  the  ropes  are  geared  di- 
rect from  the  engine  to  the  various  floors  of  the  mill,  there 
is  frequently  a  heavy  stress  on  the  shaft,  especially  on  the 
upper  floors,  due  to  the  weight  of  the  ropes:  under  such 

such  comparatively  rough  and  imperfectly  lubricated  bearings  as  we 
have  been  considering  in  which  the  friction  between  the  rubbing 
surfaces  in  contact  and  not  the  viscosity  of  the  lubricant  is  a  measure 
of  the  resistance.  See  paper  by  Prof.  Denton  in  American  MacJiinist^ 
Oct.  23,  1890, 


156 


ROPE-DRIVIKG. 


conditions  the  shafting  should  be  considered  as  head-shafts. 
In  work  of  this  nature  the  velocity  of  the  ropes  is  usually 
much  greater  both  from  the  engine  to  the  first  shaft,  and 
from  the  latter  to  the  machine  or  secondary  shaft.  Assum- 
ing a  speed  of  rope  double  that  used  for  the  previous  tables, 
and  letting  the  weight  of  pulleys,  ropes,  clutches,  and  coup- 
lings equal  four  times  the  weight  of  shaft,  it  can  be  shown 
that  the  formula  for  the  friction  load  will  be  reduced  to 


0.0057<f  A']2  -f  (0.033flLVr)'. 

TABLE  XVIII. — POWER  ABSORBED  BY  HEAD-SHAFTS  CARRYING 
HIGH-SPEED  ROPES. 


Diameter  of 
Shaft  in  inches. 


Revolutions 


Percentage  of  Loss  when  Length  of 
Shaft  in  Feet  = 


" 

50 

100 

100 

4.2 

8.1 

2* 

250 

4.8 

8.6 

. 

400 

5.6 

9.2 

100 

4.3 

8.2 

3 

^250 

5.2 

8.7 

(400 

6.0 

9.5 

100 

4.3 

8.3 

Si 

2oO 

5.3 

8.8 

400 

6.2 

9.6 

100 

4.4 

8.3 

4 

250 

5.6 

9.1 

400 

7.1 

10.1 

Line  of  action  of  resultant  of  tensions  in  main  drive  =  30°. 
Velocity  of  ropes  from  engine  5280  feet  per  minute. 
Ropes  from  shaft  are  horizontal,  and  run  at  an  average  of  2640  feet 
per  minute. 

All  ropes  drive  from  one  side  of  the  shaft  toward  the  engine. 
Weight  on  bearings  four  times  weight  of  bare  shaft. 

From  this  formula  Table  XVIII  has  been  calculated, 
and  may  be  considered  to  represent  the  percentage  of  loss 
in  the  first  shaft  when  working  under  full  load;  with 
lighter  loads  the  percentage  will  be  greater. 

It  must  be  noted  that  with  any  other  arrangement  of  shafts 
the  friction  loss  will  vary.  In  the  present  case  the  ropes 
from  the  engine  to  the  jack-shaft  make  an  angle  of  25°  with 


ROPE-DRIVING.  157 

the  horizontal,  and  the  ropes  from  the  jack-shaft  back  to 
the  machine  or  secondary  shaft  are  horizontal,  as  in  Fig.  54. 

When  rope- wells  are  used,  each  successively  higher  shaft 
will  have  an  increased  friction  percentage,  since  the  belt- 
pull  becomes  more  nearly  vertical,  and  the  resultant  load 
on  each  shaft  is  thereby  increased. 

It  is  worthy  of  remark  that  in  long  lines  of  shafting 
with  high  rim  velocity  the  influence  of  belt-pull  on  the 
bearings  is  very  slight  compared  to  the  weight  of  shaft 
and  pulleys,  so  that  the  loss  in  friction  is  but  little  more 
than  that  due  to  weight  alone.  We  see  in  this  an  addi- 
tional argument  for  high  rotative  speeds  in  shafting,  for, 
while  the  percentage  of  loss  increases,  from  8.1  to  9.2  in  a 
2£-inch  shaft  100  feet  long  running  at  100  and  400  revolu- 
tions per  minute  respectively,  the  power  transmitted  by  the 

d*N . 

shaft,  as  calculated  from  HP  =  -— V  increases  from  21^  for 

125   • 

100  revolutions  per  minute  to  about  87  h.  p.  for  400  revolu- 
tions per  minute,  so  that  while  the  friction  percentage 

9.2 
increases  in  the  ratio  ^  -  =  1.14,  the  power  transmitted 

o.  I 

increases  in  the  ratio  of  the  number  of  revolutions  per 
minute,  or  4  to  1.  In  the  first  case  the  friction  loss  is 
21-J  X  .081  =  1.74  h.  p.;  and  in  the  second,  the  loss  is 
87  X  .092  =  8.00;  therefore  the  net  power  transmitted  by 
the  shaft  running  at  100  revolutions  per  minute  is  19  h.  p. 
whereas  by  increasing  the  speed  to  400  revolutions  per 
minute  the  net  power  transmitted  will  be  79  h.  p.  With 
higher  belt  velocities  and  increased  rotative  speeds  in  our 
factory  shafting  the  friction  loss,  instead  of  being  from  30 
to  50  per  cent  of  the  total  power  transmitted,  ought  not  to 
exceed  one-half  of  these  percentages;  for  with  higher  speeds 
narrower  and  lighter  pulleys  could  be  used,  the  belts  could 
be  run  slacker,  and  lighter  shafting  could  be  employed. 
Although  it  has  been  previously  considered  in  a  general 


158  ROPE-DRIVING. 

way  that  the  friction  due  to  the  shafting  may  be  taken  as 
about  10  per  cent  of  the  full  load  transmitted  to  the  shaft, 
yet,  in  the  light  of  further  investigation,  it  will  be  seen 
that,  owing  to  the  various  conditions  under  which  the 
shafting  is  run  no  general  value  for  friction  loss  can  be 
assigned. 

While  the  friction  absorbed  by  large  engines  is  reason- 
ably less  for  rope-geared  fly-wheels  when  compared  with 
engines  using  flat  belts,  driving  in  a.similar  manner,  there 
seems  to  be  no  good  reason  for  supposing  that  the  friction 
in  ordinary  mill-shafting  should  be  appreciably  different 
when  rope-driven. 

On  account  of  the  larger  diameter  of  pulleys  used  with 
rope-driving,  the  velocity  of  the  rope  may  be,  and  usually 
is,  greater  than  that  in  a  belt  used  in  the  same  place,  and 
for  this  reason  the  pull  on  the  shaft  due  to  the  tensions  in 
the  rope  may  be  less;  but  with  long  lines  of  rope-driven 
mill-shafting  the  main  drives  only  are  of  rope,  and  any 
difference  in  pull  on  the  bearings  which  might  exist  in 
favor  of  the  rope-driven  main  shaft  must  necessarily  be 
small  when  compared  with  the  total  friction  load  due  to 
the  pull  of  the  numerous  cross  or  machine  belts  which, 
running  at  a  greatly  reduced  speed,  produce  by  far  the 
greater  effect  on  the  shaft.  With  short  lines  of  shafting, 
however,  there  will  generally  be  a  small  saving  in  favor  of 
a  rope-driven  plant.  Under  these  conditions  the  effect  of 
the  ropes  to  and  from  the  first  motion  shaft  is  usually  in 
excess  of  that  due  to  the  belts  which  may  be  used  to  trans- 
mit power  from  the  main  or  jack  shaft  to  secondary  shafts 
or  machines;  and  therefore,  since  the  rope-pull  is  less  than 
would  be  produced  by  belts  used  in  the  same  place,  we 
may  expect  the  friction  to  be  less.  When  ropes  are  used 
entirely,  as  in  electric  and  other  power  stations,  we  should 
expect  the  friction  loss  to  be  somewhat  less,  assuming  that 
the  ropes  are  run  at  a  higher  speed  than  would  be  used 
for  belts  in  the  same  place. 


KOPE-DRIVIXG.  159 


CHAPTEK  XL 

IT  has  been  stated  that  a  further  decrease  of  from  5  to 
8  per  cent  of  the  power  transmitted  by  a  rope  may  be  at- 
tributed to  losses  in  the  rope  itself  due  to  resistance  to 
bending,  wedging  in  the  groove,  differential  driving  effect, 
and  creep,  all  of  which  affect  the  loss  to  a  greater  or  lesser 
extent. 

Various  formulas  have  been  proposed  by  several  eminent 
authorities  by  which  the  resistance  of  a  rope  to  bending 
might  be  determined.  Eytelwein's  formula  assumes  that 
the  resistance  of  a  rope  is  directly  proportional  to  the  ten- 
sion and  the  square  of  the  diameter,  and  inversely  propor- 
tional to  the  radius  of  curvature  of  the  pulley;  in  which 
case  the  stiffness  of  a  hemp  rope  for  each  winding  and  un- 
winding is  given  by 


where  c  is  a  constant  equal  to  0.23,  d  is  the  diameter  of 
rope  in  inches,  r  is  the  radius  of  pulley  in  inches,  and  T 
is  the  tension  in  the  rope.  If  the  ratio  of  the  diameter  of 
rope  to  the  diameter  of  pulley  over  which  it  runs  equals  1 
to  30,  the  above  formula  becomes,  for  a  rope  running  over 
two  pulleys, 

a-  =  O.QSdT. 

Reuleaux  states  that,  since  transmission-ropes  are  usu- 
ally quite  slack,  the  coefficient  of  stiffness  should  be  taken 
somewhat  less  than  Eytelwein's  value,  and  suggests  that 


1  60  KOPE-DKIVING. 

two-thirds  would  represent  a  fair   approximation;    this 
would  give 

2  fP  d* 
o-=~X  0.23-  T  =  0.15-  T 

3  r  r 

for  each  pulley  in  the  system. 

If  the  tension  on  the  tight  and  slack  sides  of  the  rope  be 
represented  by  Tl  and  T^  respectively,  the  average  load 
on  the  rope  may  be  considered  equal  to  \(T^-\-T^\  if, 
further,  the  conditions  be  assumed  such  that  the  slack-side 
tension  equals  one  half  that  in  the  tight  or  driving  side,  — 
we  shall  have  T—  J(f  T7,).  Hence  for  two  pulleys,  when  the 
diameter  of  the  latter  equals  30  times  the  diameter  of  rope, 


=  0.02^x17;. 

Since  2",  has  been  taken  in  our  previous  work  as  equal 
to  200d2  pounds,  the  stiffness  in  the  rope  will  now  become 

X  150<P 


Now  under  these  relations  of  tension  the  driving  force 
may  be  obtained  from 


hence  the  ratio  of  loss  due  to  bending  will  be 


For  a  f-inch  rope  running  over  two  pulleys  the  loss 
equals  2.25  per  cent,  while  for  a  2-inch  rope  under  similar 
conditions  the  loss  becomes  6  per  cent.  This  is  what  we 


HOPE-DRIVING.  161 

might  expect;  for  it  is  reasonable  to  suppose  that  the  per- 
centage of  loss  should  increase  with  the  diameter  of  rope. 

It  will  be  noticed  that  the  work  done  in  bending  a  rope 
over  its  pulley  is  directly  proportional  to  the  number  of 
bends,  and  therefore  in  designing  a  rope  transmission 
every  effort  should  be  made  to  restrict  the  number  of 
bends,  as  this  is  not  only  a  large  factor  in  the  wear,  but,  as 
just  shown,  the  power  transmitted  for  a  given  tension  is 
constantly  reduced  as  the  number  of  bends  increases. 

This  feature  is  a  decided  disadvantage  with  installations 
on  the  continuous-rope  system  where  one  rope  is  bent  in 
both  directions  around  a  number  of  pulleys  on  the  several 
floors  of  a  factory.  Under  such  conditions,  and  also  in  those 
special  cases  where  it  is  absolutely  necessary  to  run  pulleys 
smaller  than  that  obtained  from  the  formula*  D  =  d17  X 


2",  the  user  should  put  in  the  very  best  quality 
of  loosely  twisted  rope  and  run  it  at  a  less  strain  than 
would  otherwise  be  adopted;  for  under  such  conditions  the 
flexibility  and  elasticity  of  the  rope  are  more  desirable  than 
a  high  breaking  strength. 

While  the  foregoing  formula  of  Eytelwein  may  give  a 
measure  of  the  force  required  to  bend  a  rope  over  a 
pulley  under  a  certain  set  of  conditions,  it  will  be 
evident,  since  the  conditions  vary  considerably  in  dif- 
ferent installations,  that,  in  order  to  be  generally  appli- 
cable to  any  given  case,  a  formula  must  contain  other 
factors  than  those  included  in  Eytelwein's  and  other  simi- 
lar formulas.  In  a  flying  rope  running  in  V  grooves,  in 
addition  to  the  bending  of  the  fibres  there  is  a  permanent 
reduction  in  cross-section,  due  to  the  uniform  compression 
of  the  rope,  as  will  be  noticed  by  measuring  a  rope  that 
has  been  running  some  time;  besides  this  there  is  a  tem- 
porary deformation  due  to  the  distortion  of  the  rope  as  it 
passes  over  the  pulley. 

*  See  page  179. 


162  KOPE-DRIVIXG. 

The  flexibility  and  elasticity  of  the  rope  undoubtedly 
have  much  to  do  with  the  resistance  to  bending,  and  the 
degree  of  twist  put  into  a  rope  is  an  important  factor  in 
tliis  connection;  for  although  hard-twisted  ropes  are  very 
much  stronger  than  those  loosely  twisted,  the  internal 
wear  is  much  greater,  as  the  fibres  are  held  more  rigidly 
and  do  not  slide  as  freely  upon  each  other.  The  advan- 
tages of  lubricating  the  fibres  of  a  manilla  rope  have  been 
already  discussed;  it  is  sufficient  to  state  here  that  the 
degree  of  lubrication  affects  the  flexibility  of  the  rope,  and 
hence  enters  as  a  factor  in  determining  the  loss  due  to 
bending.  The  varying  angle  of  contact  and,  to  a  lesser 
extent,  the  angle  of  the  groove,  must  also  have  a  certain 
influence  upon  the  resistance.  In  view  of  these  considera- 
tions it  will  be  seen  that  any  deductions  from  existing  for- 
mulae are  of  doubtful  utility  when  applied  to  transmission- 
ropes  in  use,  and  shoult?  be  considered  only  as  relative,  and 
not  absolute. 

Another  source  of  loss  is  that  due  to  the  wedging  of 
the  rope  in  the  groove.  Although  this  action  exists  to  a 
greater  or  less  extent  in  all  rope  transmissions  where  the 
shape  of  the  groove  is  such  that  the  rope  does  not  bottom, 
yet  it  is  undoubtedly  true  that  its  effect  in  a  well-con- 
structed plant  has  generally  been  over-estimated.  That  it 
does  exist  to  a  harmful  degree  can  be  seen  in  many  instal- 
lations from  the  way  in  which  the  tight  side  of  the  rope 
follows  upon  the  driven  pulley.  With  the  single-rope 
method  this  is  especially  true  in  new  installations,  where 
the  tension  is  purposely  made  rather  high  to  allow  for 
stretch  in  the  rope;  it  is,  however,  frequently  found  in  the 
continuous-rope  system  where  the  tension-carriage  is  over- 
weighted. The  factors  which  enter  into  the  consideration 
of  this  loss  are:  Tension  on  the  driving  and  slack  sides  of 
the  rope,  the  angle  of  groove,  and  the  velocity,  weight, 
and  condition  of  the  ropes.  As  we  have  previously  shown 


(page  132),  the  driving-side  tension  T:  is  made  up  of  three 
parts,  namely,  the  driving  force  P,  the  centrifugal  force 
FQi  and  the  tension  jT3,  necessary  to  balance  the  strain  for 
adhesion;  that  is, 


In  like  manner  the  tension  in  the  slack  side  of  the  rope 
consists  of  the  strain  necessary  for  adhesion  plus  the 
strain  due  to  centrifugal  force,  that  is, 

Tt  =  T      F. 


It  has  been  claimed  that  no  loss  can  occur  in  pulling  the 
rope  out  of  the  groove,  since  the  centrifugal  force  set  up 
in  the  rope  is  many  times  greater  than  any  caused  by  the 
tension  on  the  slack  side  when  leaving  the  pulley;  but  it 
is  obvious  that  a  part  cannot  be  greater  than  the  whole, 
and  therefore  the  centrifugal  force,  while  greatly  reducing 
the  wedging  force,  cannot  altogether  eliminate  it. 

An  abnormal  degree  of  slack-side  tension  has  a  direct 
effect  upon  the  wedging  of  the  rope  in  the  groove,  for  if 
sufficient  sag  is  not  allowed  and  the  slack-side  tension  is, 
therefore,  needlessly  great,  it  follows  that  the  power  trans- 
mitted will  be  reduced,  since  the  driving  force,  P,  is  equal 
to  the  difference  of  the  tension  in  the  two  portions  of  the 
rope  —  Tl  —  T^  on  the  other  hand,  the  force  drawing  the 
rope  into  the  groove  will  be  increased,  since  the  force 
equals 


It   is   obvious,   therefore,   that   the   slack  -side   tension 

*  Although  the  centrifugal  force  increases  the  tension  in  the  rope, 
its  teudency  is  to  cause  the  latter  to  leave  the  pulley,  and  therefore 
it  should  not  be  considered  as  a  part  of  the  forces  drawing  the  rope 
into  the  groove. 


164  ROPE-DRIVING. 

sliould  be  no  greater  than  just  sufficient  to  give  adhesion 
to  the  ropes  and  prevent  undue  slip  at  the  desired  speed. 

As  to  the  best  angle  for  the  groove  in  the  pulley,  opinion 
is  still  somewhat  divided;  but  in  England  the  general  prac- 
tice seems  to  favor  an  angle  of  45°  as  the  most  suitable.* 

In  the  earlier  installations  a  more  acute  angle  was  used, 
as  evidenced  by  the  discussion  of  Mr.'  Durie's  paper  on 
Rope-driving,  presented  to  the  British  Institution  of  Me- 
chanical Engineers.!  Grooves  having  an  angle  of  30°  had 
been  tried,  but  it  was  found  that  the  wear  on  the  rope  was 
altogether  too  great;  40°  was  a  very  satisfactory  angle,  and 
is  still  preferred  by  many  engineers,  while  others  use  as 
large  an  angle  as  GO0.  Occasionally  half-round  grooves 
are  used;  but  with  semicircular  grooves  on  cast-iron 
pulleys  either  the  tension  in  the  rope  must  be  increased 
or  a  greater  number  of  ropes  must  be  employed:  in  any 
case  the  advantage  seems  doubtful.  With  wooden-rimmed 
pulleys,  however,  the  semicircular  groove  is  the  better 
form;  for  since  the  coefficient  of  friction  on  the  wooden 
pulley  is  from  thirty  to  fifty  per  cent  greater  than  for  a 
similarly  shaped  groove  on  an  iron  pulley,  it  follows  that 
the  tension  in  the  rope  need  not  be  so  great;  moreover, 
wooden  pulleys  that  have  been  in  use  for  some  time  would 
indicate  that  the  semicircular  groove  is  better  adapted  to 
the  work,  for  with  anything  but  very  light  loads  the  angu- 
lar groove  is  soon  cut  out  by  the  rope,  producing  a  rim 
somewhat  similar  to  the  one  shown  in  Fig.  57,  which  rep- 
resents an  angular-grooved  wooden  pulley  that  had  been  in 
use  but  a  few  months. 

With  a  semicircular  groove  in  the  first  place  the  latter 
will  retain  more  nearly  its  original  form,  and  the  wear  on 
the  rope  will  be  greatly  reduced. 

The  pliability  of  the  rope  has  considerable  influence  on 

*  M.  E.  iu  American  Macliinist,  Nov.  10,  1893. 
f  See  Proc.  lust.  M.  E.  1876. 


HOPE-DRIVING. 


165 


the  shape  of  the  groove;  while  a  30°  angle  may  give  the 
correct  shape  for  a  soft,  loosely  twisted  cotton-rope,  a  harder 
twist  may  require  an  angle  of  40°  or  even  50°;  in  the  same 
way  a  40°  groove  may  be  all  right  for  some  makes  of  ma- 
nilla  rope,  while  others  of  a  less  yielding  nature  would  give 
better  results  if  an  angle  of  50°  or  even  60°  were  used. 

At  the  present  time  there  are  very  few  pulleys  used  in 
this  country,  except  for  machine  bands,  having  grooves 
with  an  angle  less  than  40°.  Formerly  the  Yale  &  Towne 


Elec.  World 

FIG.  57. — RIM  OF  WOODEN  PULLEY  SHOWING  WEAK. 

Mfg.  Co.  employed  grooves  of  30°  for  small  cotton  ropes 
(about  f  inch  in  diameter)  driving  their  travelling-cranes; 
but  these  were  subsequently  changed  to  50°,  and  at  the 
same  time  larger  ropes,  1£  inch,  were  adopted. 

For  manilla  rope-drives  one  large  manufacturer  uses  an 
angle  of  60°  on  all  his  pulleys,  whether  of  wood  or  iron ; 
this  angle  was  arrived  at  after  much  trial,  and  represents  an 
experience  with  manilla-rope  transmission,  covering  a  great 
many  years. 

The  most  suitable  angle  of  groove  is  that  which  affords 
the  greatest  frictional  adhesion  without  undue  slip,  and  at 
the  same  time  offers  the  least  resistance  to  the  rope  in  leav- 
ing the  groove:  with  much  slip  the  rope  is  rapidly  worn 
out,  while  with  an  excessive  grip  the  wear  is  also  rapid,  and 
a  relatively  large  amount  of  force  is  absorbed  in  overcoming 
the  wedging.  The  usual  practice  in  this  country  for  both 
cotton  and  manilla  ropes  is  an  angle  of  45°;  in  many  cases 


166 


ROPE-DRIVIXG. 


FIG.  58. 


the  section  of  the  groove  is  formed  by  arcs  of  circles,  hav- 
ing a  radius  equal  to  from  3  to  4  diameters  of  rope,  in  which 
case  the  included  angle  is  constantly  changing,  and  the  co- 
efficient of  friction,  and  hence  the  grip,  will  vary  with  the 
diameter  of  rope  used.  Thus  in  Fig.  58,  if  AB  and  A'E' 

represent  two 
ropes  of  slightly 
different  diame- 
ters running  over 
a  grooved  pulley, 
the  one  which 
sinks  deeper  into 
the  groove  will 
include  tho 
greater  angle  of 
contact  ABC,i\,\\(\. 
hence  the  lesser 
coefficient  of  friction.  This  will  hold  true  of  two  ropes  hav- 
ing the  same  diameter  but  different  degrees  of  twist:  the 
harder  rope  will  not  sink  as  deeply  into  the  groove,  and  its 
coefficient  of  friction  will  be  greater  than  that  of  the  smaller 
rope,  other  things  being  equal,  on  account  of  the  lesser 
angle  in  the  groove  at  the  point  of  tangency  of  the  harder 
rope. 

With  the  excellent  ropes  that  are  now  being  made  for 
transmission  purposes  this  form  of  groove  possesses  many 
advantages,  even  with  the  continuous- wound  system. 

In  this  system  the  tension  is  assumed  to  be  the  same  in 
each  wrap;  but  there  is  unquestionably  a  variation  of  pull 
due  to  momentary  fluctuations,  which  must  either  be  ab- 
sorbed by  the  elasticity  of  the  rope  or  transmitted  through 
the  rope  until  the  strains  are  equalized.  If  the  rope  used 
is  uniform  in  structure,  that  portion  which  receives  the 
greater  strain  will  be  drawn  more  deeply  into  the  groove; 
and  if  the  latter  be  of  the  curved  form,  the  coefficient  of 


ROPE-DKIVItfG.  167 

friction  will  be  reduced:  so  that  the  resultant  adhesion  pro- 
duced in  this  portion  of  the  rope  will  be  less  than  that  in 
some  other  part  which,  under  a  lighter  strain,  occupies  a 
position  on  the  pulley  such  that  its  coefficient  of  friction, 
and  consequent  adhesion,  is  greater  for  a  given  back-tension. 
With  the  usual  arrangement  of  pulleys,  namely,  that  in 
whicli  the  larger  wheel  is  the  driver,  the  tendency  of  an 
increase  in  tension  is  to  increase  the  velocity  of  the  driven 
pulley.  If  several  wraps  of  a  series  occupy  positions  in  the 
grooves  such  that  with  a  given  ratio  of  pulley  diameters 
the  velocity  of  the  smaller  pulley  is  twice  that  of  the  driver, 
any  decrement,  x,  of  the  eifective  radii  of  another  wrap, 

which  is  drawn  more  deeply  into  the  grooves  of  the  two 

o        o />. 

pulleys,  will  alter  the  velocity  ratio  from  T  to  -  —  ,  a  quan- 

J_  JL  —  OC1 

tity  which  must  of  necessity  be  greater  than  2.  The  ten- 
dency of  this  wrap  then  is  to  produce  a  greater  velocity  in 
the  driven  pulley,  which  cannot  occur  without  some  slip  in 
the  other  wraps;  but  these  wraps  have  a  greater  adhesion, 
and  therefore  tend  to  drive  the  pulley  at  a  less  number  of 
turns  per  minute,  which  will  produce  slip  in  the  more 
heavily  strained  member:  hence  the  effect  of  any  change 
in  position  due  to  sudden  increased  strain  on  one  wrap 
will  tend  to  quickly  adjust  the  tensions  in  all  positions  of 
the  rope  and  neutralize  any  inequalities  in  driving  effort. 
This  form  of  groove,  as  we  shall  show  subsequently,  is  even 
more  desirable  for  the  individual  rope  system,  where  the 
evils  of  differential  driving  are  frequently  so  pronounced. 
As  noted  previously,  the  frictional  grip  depends  both 
upon  the  arc  of  contact  of  the  rope  with  the  pulley  and 
the  coefficient  of  friction,  which  latter  varies  with  the  angle 
of  the  groove.  In  order,  therefore,  to  produce  the  same 
friction  on  each  pulley,  the  product  of  the  arcs  of  contact 
by  their  respective  coefficients  of  friction  must  be  equal. 
If,  as  before,  we  take  the  coefficient  of  friction  of  a  well- 


168  ROPE-DRIVING. 

lubricated  rope  on  a  smooth,  flat  metal  pulley  equal  to  0.12, 

the  coefficient  for  the  same  rope  in  a  groove  whose  angle 

/j 
is  6  degrees  will  be  0  =  0.12  cosec  —  ;  hence  with  arcs  of 

contact  a  and  a'  we  should  have  for  an  equal  grip  on  each 
pulley 

(pa  =  0'or', 

n  nj 

0.12  cosec  -  x  a  =  0. 12  cosec  —  X  a', 

&  £ 

assuming  that  the  multiplier  0.12  will  give  the  correct  co- 
efficient of  friction  on  each  pulley,  since  the  percentage  of 
slip  is  to  be  the  same. 

As  the  numerical  value  of  the  cosecant  of  an  angle  varies 
inversely  with  the  angle,  it  will  be  obvious  that  the  pulley, 
having  the  lesser  arc  of  contact,  should  also  have  the  more 
acute  angle  in  the  groove.  This  property  of  rope  friction 
is  frequently  taken  advantage  of  in  designing  a  plant,  and 
it  is  not  uncommon  to  find  the  grooves  in  the  large  wheel 
more  obtuse  than  those  in  the  smaller,  especially  when 
there  is  considerable  difference  in  the  diameters  of  the 
pulleys.* 

From  the  above  equation  there  is  obtained 

6  0'     a' 

cosec  —  =  cosec—  X  —  > 

6  £          (X 

in  which  0  and  a  represent  the  angle  of  groove  and  arc  of 
contact,  respectively,  on  the  larger  pulley,  and  0'  and  a' 
similar  values  for  the  smaller  pulley.  With  the  least  angle 
of  groove  equal  to  35°,  40°,  or  45°,  the  corresponding  angle 
in  the  larger  pulley  should  be  as  indicated  in  Table  XIX, 
when  the  ratio  of  the  arcs  of  contact  is  known. 

*Mr.  T.  Spencer  Miller,  of  the  Lidgerwood  Mfg.  Co.,  has  been 
granted  a  patent  (U.  S.  patent  No.  444919),  upon  the  application  of 
this  principle  to  continuous- rope  transmissions,  in  which  with  sheaves 
of  different  diameters  more  obtuse  grooves  are  given  to  a  larger  wheel. 


ROPE-DRIVING.  169 


TABLE  XIX.— ANGLE  OF  GKOOVE  FOR  EQUAL  ADHESION. 


Arc  of  contact  on  small  pulley       a' 

0.9 

0.8 

0.75 

0.7 

0.65 

0.6 

Arc  of  contact  on  large  pulley        a 

Angle  of  groove  in  large  pulley 
when  groove  in  small  pulley  —35° 

40° 

44° 

74° 

51* 

55° 

60° 

Angle  of  groove  in  large  pulley 

when  groove  in  small  pulley  =  40° 
Angle  of  groove  in  large  pulley 

45° 

50° 

54° 

58° 

64° 

70° 

when  groove  in  small  pulley  =45° 

50° 

55< 

60° 

66° 

72° 

80° 

Of  course  an  idler  or  binder  pulley  may  be  used  to  in- 
crease the  arc  of  contact  on  the  smaller  pulley,  and  thus 
maintain  an  equality  of  grip  on  each  pulley;  but  this  device 
produces  an  objectionable  reverse  bend  in  the  rope,  which 
should  be  avoided  as  much  as  possible  in  rope  transmission. 
Both  of  these  arrangements,  as  well  as  the  winder-pulley 
previously  discussed,  are  intended  to  prevent  slip  and  at 
the  same  time  obtain  the  maximum  adhesion  for  the  least 
amount  of  back-tension  without  increasing  the  losses  due 
to  wedging  in  the  grooves,  journal-friction,  and  wear  in 
the  rope. 

As  pointed  out  by  Mr.  W.  H.  Booth,*  although  there 
may  be  some  loss  and  wear  due  to  wedging  in  the  groove, 
by  far  the  greater  loss  is  occasioned  by  the  deeper  wedging 
of  one  rope  as  compared  with  another,  so  causing  them  to 
grip  upon  a  different  circumference,  in  which  case  each 
rope  tends  to  impart  a  different  velocity  to  the  driven  pulley; 
the  actual  resultant  velocity  will  be  a  mean  of  the  several 
velocities  of  the  individual  ropes,  so  that  slipping  and  wear 
of  some  or  all  of  the  ropes  must  occur,  due  to  the  differen- 
tial driving  thus  set  up. 

With  continuous  rope  transmissions  this  effect  is  not 
so  apparent,  although  it  exists  to  a  certain  extent  on  account 
of  inequalities  in  the  rope  and  various  mechanical  imper- 
fections in  the  system;  but  in  the  English  or  independent 

*  American  Machinist,  Dec.  8,  1888. 


170  HOPE-DRIVING. 

rope  transmission  its  effect  is  very  marked,  and  is  generally 
considered  as  the  principal  source  of  loss  of  efficiency  in 
this  system.* 

While  there  is  undoubtedly  a  considerable  loss  which 
may  be  charged  to  this  cause,  it  is  also  true  that  its  effect 
may  be  greatly  reduced  by  a  careful  study  of  the  require- 
ments of  the  problem  and  an  intelligent  application  of 
correct  principles  to  the  case  in  hand. 

It  is  evident  that  in  order  to  prevent  slip  and  the  loss  of 
power  incident  thereto  it  will  be  necessary  to  obtain  a  uni- 
form velocity  in  the  several  ropes  running  over  a  pair  of 
pulleys;  to  approach  this  desideratum  each  set  of  ropes 
should  be  of  the  same  make  and  degree  of  hardness,  of 
uniform  diameter,  evenly  spliced,  having  the  same  amount 
of  sag  in  both  members,  and  run  over  grooves  of  uniform 
diameter,  shape,  and  smoothness. 

Obviously  it  would  be  impossible  in  practice  to  maintain 
all  these  conditions,  even  if  it  were  practicable  to  overcome 
the  mechanical  difficulties  and  install  a  plant  under  the 
given  requirements  ;  yet  much  may  be  done  to  effect  the 
desired  end. 

Uniformity  of  length  is  an  important  requirement,  for,  if 
one  rope  of  a  set  be  allowed  a  less  amount  of  sag  than  the 
others,  the  sum  of  the  tensions  will  be  greater  in  this  rope, 
and  in  consequence  it  will  be  drawn  more  deeply  into  the 
groove;  its  pitch  diameter  will  therefore  be  reduced  and 
its  velocity  will  be  different  from  the  others  in  the  set,  in 
which  case  if  the  driving  and  driven  pulleys  are  of  unequal 
diameters  the  tendency  will  be  to  give  the  driven  pulley  a 
different  velocity,  and  slip  must  necessarily  occur. 

The  following  from  the  American  Machinist  is  pertinent 
to  the  subject: 

"It  will  be  readily  seen  that  in  a  set  of  ten  or  a  dozen 

*  American  Machinist,  Dec.  1,  1892. 


ROPE-DRIVING.  171 

ropes,  each  of  a  somewhat  different  length,  the  loss  of 
power  from  this  cause  may  easily  become  a  very  serious 
item,  and  to  this  there  is  to  be  added  the  correspondingly 
diminished  life  of  the  ropes.  A  similar  action  occurs 
when  worn  ropes  are  allowed  to  work  conjointly  with  new, 
even  though  the  deflections,  and  therefore  the  tensions,  on 
the  several  ropes  are  practically  equal.  In  this  case  the 
loss  due  to  abnormal  tension  and  wedging  of  some  of  the 
ropes  into  the  grooves  is  avoided,  but  the  differential  driving 
effect  due  to  the  ropes  virtually  running  on  pulleys  of 
different  diameters  still  exists,  and  is  equally  objectionable. 
It  may  here  be  noted  that  the  effect  of  the  differential 
driving  upon  the  ropes  depends  to  a  very  large  extent 
upon  the  relative  diameter  of  the  two  pulleys. 

"It  is  evident  that,  when  the  driving  and  driven  pulleys 
are  of  the  same  diameter,  any  variation  in  the  effective 
pitch  diameters  of  the  several  grooves  will  have  no  appre- 
ciable effect  upon  the  transmission,  provided  that  the 
diameter  and  shape  of  the  corresponding  grooves  in  the  two 
pulleys  are  the  same.  It  may  be  noted,  however,  that  the 
worn  ropes  which  run  deeper  in  the  grooves,  having  a 
slightly  less  velocity,  are  subjected  to  a  somewhat  greater 
stress  than  their  newer  and  larger  companions." 

With  a  form  of  groove  similar  to  that  shown  in  Fig.  58, 
in  which  the  sides  of  the  groove  are  circular  arcs  of  large 
radius,  there  will  be  a  tendeocy  to  correct  the  differential 
driving,  especially  so  when  the  driving  pulley  is  smaller 
than  the  driven.  In  this  case,  if  we  assume  that  all  the 
ropes  in  a  set  are  put  on  with  the  same  amount  of  sag,  so 
that  the  tensions  are  practically  equal  for  both  old  and  new 
ropes,  any  difference  in  diameter  of  rope  will  cause  the 
larger  to  carry  more  than,  its  share  of  the  load,  since  the 
effective  radius  of  the  pulley,  and  therefore  the  velocity  of 
the  rope,  is  greater;  moreover,  since  the  coefficient  of  fric- 
tion varies  inversely  with  the  depth  of  contact  in  this  form 


172 


ROPE-DRIVING. 


of  groove,  the  larger  rope  is  acted  upon  by  a  more  intense 
grip,  by  reason  of  which  more  work  is  imposed  upon  it, 
while  a  smaller  rope  will  be  relieved  of  some  of  its  work. 

The  tendency  of  the  larger  rope  is  to  turn  the  driven 
pulley  at  an  increased  velocity.     Thus  in  Fig.  59  if  D  be 


R=  -3"  '  F         •  Elec.  World 

FIG.  59. 
the  driving-pulley  and  7^  the  follower,  with  respective  diam- 

T> 

eters  such  that   the  velocity  ratio  --  is  1  to  3,  then  the 

smaller  ropes  o  will  tend  to  drive  .Fin  the  ratio  £,  while 
the  larger  ropes,  n  working  at  a  distance  x  further  away 
from  the  centre  of  pulleys,  will  tend  to  drive  .Fin  the  ratio 


,  and  therefore  at  a  higher  velocity.     The  resulting 


1  +  x 
-  -- 

O    ~T~    «*/ 

velocity  of  the  follower  will  depend  upon  the  work  done  by 
each  rope,  so  that  some  slip  must  necessarily  occur;  on 
account  of  the  greater  load  taken  by  the  larger  ropes  they 
will  be  rapidly  compressed  and  worn:  hence  any  initial 
variation  in  turning  effort  will  be  speedily  reduced  to  a 
minimum  by  this  equalizing  process,  which,  although  it 
incurs  loss,  ultimately  insures  a  better  distribution  of  the 
work  with  the  least  wear  on  the  ropes. 

On  the  other  hand,  when  the  driving-pulley  is  larger 
than  the  follower  the  smaller  ropes  are  drawn  further  into 
the  grooves  and  tend  to  impart  an  increased  velocity  to  the 
driven  'sheave.  Thus  in  Fig.  60  the  large  pulley  is  the 

7?        ^ 
driver  and  the  velocity  ratio  is  —  =  -  with  the  smaller 


ROPE-DRIVING.  173 

ropes  0;  with  the  large  ropes  n,  working  at  a  distances 
nearer  the  circumference,,  the  tendency  will  be  to  produce 

a  ratio  equal  to  z-      -  :  hence  the  effect  will  be  retardation. 
1  +  x 

Since  the  larger  ropes  acting  higher  in  the  groove  have  an 
increased  grip  and  speed  they  will  exert  a  greater  influence 
upon  the  smaller  pulley;  and  although  the  smaller  ropes 
may  be  drawn  more  deeply  into  their  grooves  in  attempt- 


FOLLOWER 

_  Elec.World 

R=3r 

FIG.  60. 

ing  to  drive  the  follower  at  a  greater  number  of  revolutions 
per  minute  their  effect  is  lessened  on  account  of  the  dimin- 
ished grip,  due  to  the  form  of  groove,  in  consequence  of 
which  the  larger  rope  acts  with  a  greater  effect  and  the 
smaller  with  a  lesser  effect  than  would  obtain  with  an  ordi- 
nary V-groove  under  like  conditions. 

To  equalize  the  driving  efforts  of  a  number  of  ropes,  and 
to  prevent  the  slip  which  must  inevitably  occur  with  a 
solid-grooved  rim,  Mr.  John  Walker  has  devised  and  pa- 
tented* a  "differential"  driving-pulley, since  it  allows  the 
ropes  to  travel  at  different  speeds  suited  to  the  conditions 
imposed  upon  each  rope. 

Originally  intended  for  cable-railway  machinery,  where 
the  wear  on  the  drums  due  to  the  wire  cable  is  excessive, 
the  differential  principle  has  been  extended  to  other  uses, 
notably  elevator  sheaves  and  rope-transmission  pulleys. 

In  the  latter  the  rope  is  led  over  a  number  of  separate 

*  Feb.  23,  1893. 


174 


ROPE-DKIVING. 


FIG.  61. 


rings,  Fig.  61,  adapted  to  turn  loosely  and  independently 

of  each  other  on  the 
smooth  circumfer- 
ence of  the  drum. 

While  the  rope  is 
passing  over  the  pul- 
ley the  tendency  of 
the  rings  will  be  to 
adjust  themselves  to 
the  strain  in  each 
member  by  moving 
around  the  circumference  of  the  drum.  Thus  the  driving- 
tension  is  equalized,  and  each  rope  is  brought  to  do  its  own 
share  of  the  work  without  slipping  in  its  groove.  These 
rings  have  a  diametrical  friction,  due  to  the  pressure  of  the 
rope  in  the  groove  transferred  to  the  flat  surface  of  the 
drum.  In  addition  to  this  an  adjustable  rubber  washer  is 
inserted  between  the  rim  of  pulley  and  loose  flange,  so  that 
by  tightening  this  adjustable  joint  the  separate  rings  are 
caused  to  exert  enough  pressure  upon  each  other  to  pro- 
duce a  certain  amount  of  friction  on  the  side  surfaces,  the 
combined  friction  of  the  several  rings  being  sufficient  to 
drive  the  pulley  or  ropes,  as  the  case  may  be.  In  this  way 
each  rope  bears  its  due  share  of  the  work,  as  the  adjust- 
ment is  such  that  the  friction  between  the  several  parts 
is  brought  into  equilibrium. 

In  practice  the  axial  rotation  of  the  ropes  will  frequently 
exert  a  modifying  influence  on  the  differential  driving, 
since,  other  conditions  being  unchanged,  a  rotating  rope 
tends  to  maintain  its  circular  form,  and  therefore  will  work 
less  deeply  into  the  groove.  An  additional  advantage  is 
that  such  rotation  promotes  the  durability  of  the  rope,  as 
the  wear  is  more  uniform. 

The  loss  due  to  the  elastic  slip  or  creep  of  the  belt  has 
some  influence  upon  the  efficiency  of  transmission,  but  in 


ROPE-DRIVING.  175 

any  case  its  effect  is  small.  When  an  elastic  body,  such 
as  a  rope,  is  placed  under  tension,  it  stretches,  and  the 
elongation,  within  the  limit  of  elasticity,  is  proportional  to 
the  strain  in  the  rope.  When  power  is  transmitted  from 
one  pulley  to  another  the  driving  side  is  subjected  to  a 
greater  tension  than  the  slack  side,  in  consequence  of  which 
the  velocity  in  the  driving  side  will  be  slightly  greater  than 
that  in  the  slack  side,  and  the  circumferential  velocities  of 
the  two  pulleys  will  not  be  the  same.  This  will  be  evi- 
denced from  the  following  considerations: 
Let  V  =  circumferential  velocity  of  driver; 

V  =  "       "  follower; 

Tj  =  tension  in  driving  side  of  rope; 

TI=       "       "  slack  side  of  rope; 

A   —  cross-section  of  rope  in  square  inches; 

L   =  original  length  of  a  piece  of  the  rope  in  either 
=      side  in  its  normal  condition; 

e     =  elongation  in  driving  side  due  to  tension  T  ; 

e'    =  l<         "  slack  side  due  to  tension  T^\ 

E  =  modulus  of  elasticity  of  the  rope. 

T  T 

When  the  rope  is  at  work  e  =  — ^  and  e'  =  -^,  so  that 

-tii  A  Mi  A 

the  length  of  each  member  will  now  be  L  -j-  e  and  L  -j-  e', 
respectively.  The  length  of  rope  running  on  to  the  driving- 
pulley  in  a  unit  of  time  will  therefore  be  greater  than  that 

delivered  to  the  driven  pulley  in  the  proportion  y-^—  „  and 

L  -j-  e 

the  velocity  of  the  two  pulleys  will  now  be 

^|±i;    hence  T<=  Kx£±f 

T  T 

Calling  L  unity,  and  assuming  —j f  =  2,  -±  =  320  pounds 

•*«  -« 

(since   T^  =  200d2  pounds,  and  A  =  0.8  Xj^2),  and  E  — 


176  ROPE-DKIVING. 

40000,*  we  have 

320  ,  _       320 

~  40000  ~  2X40000; 

therefore 

'A  =  0.996F; 


L  +  40000 

that  is,  the  loss  due  to  creep  when  all  the  ropes  are  working 
under  normal  conditions  is  about  one  half  of  one  per  cent. 
With  different  ratios  of  driving  to  tight-side  tension  and 
different  intensities  of  stress  it  is  evident  that  this  loss  will 
vary  in  the  different  ropes;  in  any  case  the  loss  ought  not 
to  exceed  one  and  a  half  per  cent,  as  will  be  seen  if  we  as- 
sume extreme  conditions. 

For  instance,  let  the  stress  on  the  rope  be  600  pounds 
per  square  inch  of  section  and  assume  the  ratio  of  tensions 
equals  one  to  five;  with  the  most  elastic  long-staple  cotton 
rope  it  is  possible  that  the  modulus  of  elasticity  may  be  re- 
duced to  30,000  pounds;  therefore,  under  these  conditions, 
the  velocity  of  the  circumference  of  the  driven  pulley  will 
be 


30000 

thus  representing  a  loss  of  1.6  per  cent.  It  is  probable 
that  one  per  cent  is  an  ample  allowance,  even  under  un- 
favorable conditions. 

*  See  Reports  of  Tests  on  Manilla  Bope,  Ex.  Doc.  No.  36,  1885. 


177 


CHAPTER  XII. 

THE  construction  of  rope  pulleys  is  a  matter  of  consider- 
able importance,  for  it  is  evident  from  what  has  preceded 
that  the  size,  shape,  and  condition  of  the  pulley  all  exert  a 
marked  influence  upon  the  efficiency  of  rope  transmission. 

It  has  been  shown  that  the  size  of  the  pulley  materially 
affects  the  wear  of  the  rope:  the  larger  the  sheaves  the  less 
the  fibres  of  the  rope  are  flexed,  and  the  less  they  slide  on 
each  other;  consequently  there  is  less  internal  wear  of  the 
rope. 

The  minimum  diameter  of  the  pulley,  as  given  by  differ- 
ent authorities,  varies  from  thirty  to  forty  times  the  diame- 
ter of  rope  to  be  used.  Forty  times  the  diameter  of  rope 
for  manilla  is  excellent  practice,  and  experience  has  shown 
that  a  larger  multiplier  would  be  still  better,  as  the  larger  a 
pulley,  the  better  for  either  belt  or  rope  passing  over  it; 
but  such  a  rule,  although  convenient  to  use,  was  evidently 
founded  upon  largo-sized  ropes  running  at  a  high  velocity, 
and  a  little  consideration  will  show  that  while  forty  times 
the  diameter  of  rope  may  be  all  right  for  a  two- inch  rope, 
it  is  also  true  that  a  lesser  proportion  will  give  suitable  di- 
ameters of  pulley  for  a  one  inch  rope.  If  we  take  two  ropes 
dj  and  J2 ,  whose  diameters  are  one  and  two  inches,  respec- 
tively, and  bend  them  over  pulleys  of  the  same  diameter, 
the  fibres  in  dl  will  be  stressed  an  amount  equal  to  x  due  to 
the  stretch  and  sliding  of  the  fibres  one  upon  another.  On 
the  other  hand,  since  the  outer  fibres  in  c/2  are  twice  as  far 
from  the  neutral  axis,  these  fibres  will  stretch  and  slide 
upon  each  other  to  an  extent  equal  to  twice  that  produced 


178  ROPE-DRIVIKG. 

in  the  smaller  rope  ;  but  the  relation  of  the  areas  of  the 
two  ropes  is 


under  the  given  conditions  ;  hence  the  total  slip  in  the 
larger  rope,  due  to  the  sliding  of  the  fibres  upon  each  other, 
will  be  eight  times  greater  than  in  the  smaller  one,  or  as 

-y-f.    Therefore,  for  equal  extension  and  slip  of  the  fibres  in 

bending  a  rope  over  a  stationary  pulley,  it  would  appear 
that  the  pulleys  should  have  diameters  proportional  to  the 
cube  of  the  diameters  of  the  ropes. 

In  the  above  consideration  no  account  has  been  taken  of 
the  external  wear,  produced  by  slipping  in  the  groove, 
wedging,  rubbing  contact,  and  other  causes  of  loss  which 
affect  a  running  rope;  for  a  given  velocity  these  losses  will 
increase  with  an  increased  speed  of  rotation,  and  hence  will 
be  greater  with  a  smaller  pulley,  which  would  indicate  that 
the  diameter  of  the  latter  should  not  be  directly  propor- 
tional to  the  cube  of  the  diameter  of  the  rope.  It  is  also 
evident  that  a  rope  running  at  5000  feet  per  minute  will 
be  subjected  to  a  greater  number  of  bends  and  a  greater 
external  wear  than  it  would  if  running  over  the  same  pul- 
leys at  2000  feet  per  minute.  The  first  of  these  influences 
is  recognized  by  many  engineers  who  use  a  rule  approxi- 
mating the  following: 

For  the  least  diameter  of  pulley,  D,  multiply  the  circum- 
ference of  rope  by  ten  times  its  diameter  and  divide  by 
two.  *  This  is  practically  equal  to  D  =  15tT  inches. 

While  we  believe  in  using  as  large  a  pulley  as  possible  in 

*Mr.  Jas.  Gamble  in  Textile  Recorder, 


HOPE-DRIVING.  179 

any  given  case,  conditions  will  arise  when  it  is  desirable  to 
use  the  smallest  possible  diameter  without  excessive  injury 
to  the  rope.  In  such  case  the  individual  circumstances 
should  be  considered  by  the  designer. 

From  the  foregoing  it  is  obvious  that  for  a  given  rope 
and  tension  the  least  diameter  of  pulley  which  may  satis- 
factorily be  used  under  known  conditions  should  be  de- 
pendent upon  the  velocity  of  the  rope,  and  should  vary 
with  the  size  of  the  latter  in  such  a  manner  that  for  any 
given  speed  the  pulley  diameter  will  be  proportional  to 
some  power,  greater  than  unity  and  less  than  the  cube  of 
the  diameter  of  the  rope. 

From  an  investigation  of  numerous  examples  in  operation 
under  varying  conditions,  some  of  which  work  satisfactorily 
and  others  very  poorly,  it  would  seem  that  while  the  value 
15d2  may  give  a  suitable  diameter  of  pulley  for  a  soft  cotton 
rope  of  small  diameter,  such  a  value  is  entirely  too  small 
for  an  equal-sized  manilla  rope;  but,  on  the  other  hand, 
40d  is  somewhat  larger  than  absolutely  necessary  for  these 
ropes,  although  it  is  always  desirable  to  use  such  a  pulley 
if  conditions  permit. 

If  for  any  reason  it  is  necessary  to  adopt  a  small  pulley, 
the  least  pitch  diameter  for  a  sheave  to  be  used  with 
manilla  rope  working  under  an  assumed  tension  of  20(k/2 
pounds  may  be  determined  from  the  following  empirical 
formula,  which  is  believed  to  represent  the  requirements  of 
good  practice: 

D  =  d1-7  X  &V+  12", 


in  which  D  =  pitch  diameter  of  pulley  in  inches; 

d  =  diameter  of  rope  in  inches; 

V=  velocity  of  rope  in  feet  per  minute. 
In  order  to  simplify  the  use  of  this  formula  the  follow- 
ing values  of  d1-7  and  ^Fhave  been  calculated  as  given 
in  Table  XX, 


180 


ROPE-DRIVING. 


Table  XXI  gives  values  of  D  for  ropes  varying  from  f 
inch  to  2  inches  in  diameter,  running  at  2000  to  5000  feet 
per  minute.  When  the  speed  of  the  rope  is  not  known  the 
diameter  given  in  the  last  column  should  be  used  for  the 
minimum  size  of  pulley;  in  fact  it  would  be  better  to  use 
these  diameters,  or  even  larger  ones,  in  all  cases,  provided 
the  constructive  features  in  the  plant  will  permit  their  ap- 
plication. With  an  increased  tension  in  the  ropes  the  diam- 
eter of  pulley  should  be  increased  also.  In  the  same  way 
if  the  working  tension  should  be  less  than  200^2  pounds, 
then  the  diameter  of  pulley  may  be  less  than  here  given. 
If  cotton  rope  be  used,  the  least  diameter  of  pulley  may 
also  be  taken  somewhat  less. 


TABLE  XX.  -VALUES  OF  d>*  AND 


D  i  a  of  ro  pe  d      

3 

1 

U 

U 

If 

2 

Value  of  d1-7          

0  61 

1 

1  46 

1  99 

2.59 

3.25 

Velocity  of  rope,  V,  in  feet 

1000 

1500 

2000 

3000 

4000 

5000 

Value  of   y'F  

10 

11  4 

12.6 

14.4 

15.87 

17  1 

TABLE  XXI.— LEAST  DIAMETEK  OF  PULLEY  FOR  GIVEN  DIAME- 
TER AND  SPEED  OF  MANILLA  ROPE. 


Diameter 

nf 

Rope. 

2000 

3000 

4000 

5000 

3 

20 

21 

22 

224. 

1^ 

251 

264 

28 

29 

U 

33 

35 

37 

u 

37 

404 

43^ 

46 

if 

44 

49 

584 

564 

2 

53 

58| 

634, 

67 

181 


TABLE  XXII.  —  ROPE  PULLEYS  FOII  GENEKAL  WORK. 


Diameter  of  Rope.  . 

I 

1 

H 

n 

If 

2 

Diameter  of  Pulley 

24 

36 

48 

60 

72 

84 

It  is  well  to  remember  that  the  cause  of  many  failures 
and  much  trouble  experienced  in  rope-driving  is  due  to 
the  use  of  too  small  a  pulley  for  the  size  of  rope  and  ten- 
sion carried,  but  we  have  yet  to  hear  of  a  case  where  the 
diameter  of  pulley  has  been  too  great.  When  it  is  not  ab- 
solutely necessary  to  restrict  the  diameter  to  the  smallest 
possible  which  may  be  used,  the  least  diameter  should  con- 
form to  that  given  in  Table  XXII,  which  has  been  arranged 
for  general  work,  and  gives  least  diameters,  to  the  nearest 
half  foot,  for  rope-pulleys  suitable  for  all  speeds  within  the 
limits  of  good  practice. 

When  a  close  velocity  ratio  between  driver  and  follower, 
greater  or  less  than  unity,  is  required,  the  pitch  diameter 
of  each  pulley  should  be  measured  from  the  point  of  tan- 
gency  of  the  rope  in  the  groove,  and  not  from  the  centre 
of  the  rope.  For,  if  D  represent  the  diameter  of  driver 
measured  to  centre  of  rope  (Fig.  62),  and  F  the  corre- 
sponding diameter  of  follower,  the  velocity  ratio  will  be 

-JFJ —  — ,  where  x  is  the  common  vertical  distance  between 

£*    —   &X 

centre  of  rope  and  point  of  contact.  Where  the  pulleys 
are  the  same  size  it  is  obvious  that  the  velocity  ratio  will 
not  be  changed,  but  as  the  difference  in  diameter  increases 
the  influence  of  x  will  be  more  marked;  thus  if  D  =  3F, 

the  velocity  ratio  will  be  -^ — — , — a  result   manifestly 

*\  Ti1 

greater  than  -.     In  any  case,  the  smaller  the  diameter  of 


182 


fcOPE-DRIVIKG. 


pulleys  for  a  given  velocity  ratio,  the  greater  the  effect  of 
the  quantity  x. 

In  ordinary  single  transmissions  it  will  be  sufficiently 
close  to  assume  the  diameter  of  pulley  as  measured  from 
centre  of  rope. 

APPARENT  DIAMETER=D 
REAL.  "          =0-2* 


FIG.  62. — DIAMETER  OF  PULLEY. 

It  is  evident  irom  the  previous  considerations  that  the 
condition  and  shape  of  the  groove  is  a  matter  of  much 
importance:  so  generally  is  this  recognized,  that  manufac- 
turers now  almost  universally  turn  their  rims  to  special 
gauges  and  templets,  which  insure  uniformity  in  diameter 
and  shape  of  groove.  Special  tools  have  been  devised  for 
turning  the  grooves,  and  one  prominent  manufacturer  has 
constructed  a  special  machine  in  which  the  rims  are  milled 
out.  By  this  process  some  of  the  smaller  pulleys  are 
machined  in  one  operation — as  many  cutters  being  used  as 
there  are  grooves  to  be  milled.  Uniformity  of  pitch,  diam- 
eter, and  contour  are  thus  insured  independently  of  the 
operator.  Formerly  the  bottom  of  the  groove  was  fur- 


HOPE-DRIVING.  183 

nished  with  spikes,  or  the  sides  were  cut  into  angular  teeth 
in  order  to  prevent  the  rope  from  slipping;*  but  a  greater 
experience  with  rope-driving  has  shown  that  the  whole  sur- 
face of  the  groove  must  be  perfectly  smooth,  and  should 
be  carefully  polished  as  well  as  machined,  since  the  fibres 
of  the  rope,  if  allowed  to  rub  on  a  rough-turned  or  cast 
surface,  will  gradually  break,  fibre  by  fibre,  and  thus  give 
the  rope  a  short  life.  It  is  also  necessary  to  avoid  using 
any  pulley  with  sand  or  blow  holes  in  the  groove,  as  they 
are  very  destructive  to  a  rope:  when  blow-holes  occur,  if 
not  honeycombed  excessively,  they  should  be  filled  in  with 
lead,  or,  preferably,  Babbitt's  metal;  otherwise  the  pulley 
should  not  be  used. 

Some  rope-pulleys  are  simply  cast  and  the  rims  smoothed 
up  by  holding  a  piece  of  abrasive  material,  such  as  a  bro- 
ken emery-wheel  or  grindstone,  against  the  surface  of  the 
groove  while  the  pulley  revolves  at  a  high  speed.  While 
this  may  produce  a  smooth  surface,  such  an  expedient  is 
doubtful  economy;  for  no  matter  how  carefulty  a  multiple- 
grooved  pulley  may  be  cast,  it  is  almost  impossible  to  ob- 
tain a  rim  all  the  grooves  of  which  are  true  and  of  the  same 
diameter:  the  result  is  that,  with  such  pulleys,  the  ropes 
tend  to  vibrate  and  sway  from  side  to  side,  rubbing  against 
each  other,  and  frequently  against  side-posts,  walls,  floors, 
and  other  obstructions,  which  rapidly  destroy  the  rope. 

Attempts  have  been  made  to  produce  a  finished  surface 
in  the  groove  by  casting  the  rim  in  an  accurately  turned 
chill,  but  such  pulleys  have  not  been  as  successful  as  an- 
ticipated. A  form  of  pulley  made  by  the  Link-Belt  Engi- 
neering Co.  of  Philadelphia  consists  of  an  iron  sheave  cast 
in  rings,  some  with  and  some  without  arms  and  hub,  from 
which  a  complete  pulley  is  built  up  having  the  requisite 
number  of  arms  for  strength,  and  at  the  same  time  being 

*  Willis :  "  Principles  of  Mechanism," 


184 


ROPE-DKIVIKG. 


light  and  free  from  excessive  shrinkage  strains  liable  to  ex- 
ist in  wheels  having  light  arms  and  heavy  rims  and  hubs. 
The  sheaves  have  a  slight  projection 
on  one  side  and  corresponding  re- 
cess on  the  other,  with  bolt-holes  at 
the  circumference,  so  that  a  multiple- 
grooved  pulley  with  any  number  of 
grooves  may  be  readily  built  up  by 
bolting  the  sections  together  as  shown 


FIG.  63.— BUILT-UP  ROPE  PULLEY  FOR  LIGHT  WORK. 
in  Fig.  63.  These  wheels  are  made  from  metal  patterns  and 
moulded  in  a  three-part  iron  flask.  The  core  which  makes 
the  groove  is  continuous  and  of  green  sand,  producing  a 
very  smooth  casting.  The  groove  is  finished  with  an  em- 
ery-wheel swinging  in  a  frame  like  a  cut-off  saw,  the  final 
finish  being  given  by  the  use  of  emery  and  oil.  While  this 
process  produces  a  single-grooved  wheel  more  cheaply  than 
by  turning,  it  is  doubtful  whether  from  a  manufacturing 
standpoint  there  is  any  economy  in  the  built-up  wheel 
when  we  consider  the  degree  of  accuracy  now  obtained  in 
moulding  multiple-grooved  wheels,  and  the  consequent  re- 
duction of  labor  in  turning  them. 

Another  special  form  of  rope-driving  pulley  is  that  made 
by  Jofyn  Mnsgrave  &  Song,  Bolton,  England.  This  pulley 
(Fig.  64),  it  will  be  noticed,  is  extremely  light,  but  is  suffi- 
ciently strong  for  the  requirements.  This  lightness  is  at- 
tained by  the  use  of  steel  arms  turned  tapering,  and  firmly 
secured  to  the  rim  and  the  hub,  which  latter  is  split  in 


HOPE-DRIVING.  185 

three  segments  and  ringed  with  steel.  By  the  use  of  this 
form  of  pulley  the  shafts  are  relieved  to  a  considerahle 
extent  of  the  weight  and  consequent  friction  entailed  by 
the  ordinary  pulley,  and  there  are  no  excessive  shrinkage 
strains  to  contend  with.,  as  is  usually  the  case  with  the 


FIG.  64. — ROPE  PULLEY  WITH  TURNED  STEEL  ARMS. 

common  form  of  single-casting  pulley.  In  pulleys  of  this 
form,  where  wrougbt-iron  and  steel  rods  are  used  for  arms, 
the  ends  of  the  rods  should  be  dipped  in  acid  and  tinned 
before  setting  in  the  mould.  In  addition  to  this  the  rods 
are  frequently  headed  up  and 
The  bottom  of  the  groove  in 


186 


ROPE-DRIVIHG. 


FIG.  65. 
WOOD-PILLED  PULLEY  RIM. 


times  filled  with  wooden  blocks  dovetailed  into  a  channel 
cast  to  receive  them,  in  which  case  the  rope  runs  on  the 
bottom  and  the  shape  of  the  groove  approximates  that 

shown  in  Fig.  65;  after 
being  fitted  and  secured 
the  groove  is  trued  up  and 
turned  out  to  the  desired 
shape.  Gutta-percha,  rub- 
ber, leather,  tarred  hemp, 
and  other  materials  have 
been  used  for  the  same  pur- 
pose; but  in  the  best  mod- 
ern practice  a  smooth  cast- 
iron  surface  is  preferred  to 
any  other,  and  we  find  these 
groove  linings  confined  chiefly  to  pulleys  used  in  wire-rope 
transmissions  and  hoisting-machinery. 

Of  late  years  an  all-wood  rim  with  V  or  U  grooves  has 
been  used  to  some  extent,  as  it  makes  a  cheap  pulley,  and 
is  very  satisfactory  for  light  work  when  a  semicircular 
groove  is  adopted.  Wooden  rims,  however,  should  not  be 
used  for  heavy  work,  and  the  slip  of  the  rope  should  be  re- 
duced to  a  minimum  ;  otherwise  the  grooves  and  ropes  will 
be  rapidly  cut  out  and  the  whole  system  will  be  very  unsat- 
isfactory. 

The  proportions  of  rope-pulley  rims  depend  somewhat 
upon  the  shape  of  groove  adopted.  Some  manufacturers 
have  a  standard  groove,  with  straight  sides,  which  may  be 
used  for  several  different-sized  ropes,  as  shown  in  Fig.  66. 
For  light  work  such  a  pulley  is  very  satisfactory,  and  has 
many  advantages  in  regard  to  constructive  features  not 
possessed  by  any  other  form.  The  groove  shown  in  Fig. 
67*  is  very  commonly  used  in  England,  but  in  this  country 


Low  and  Bevis,  "  Machine  Design,"  p.  156, 


ROPE-DRIVItfG. 


187 


manufacturers  of  large  rope-pulleys  usually  prefer  a  form 
in  which  the  abrupt  change  in  profile  (as  at  a)  does  not 
occur.  Of  these  the  more  common  form  is  a  modification 


FIG.  66  — RIM  SECTIONS— STRAIGHT  GROOVE. 

of  the  English  section,  in  which  the  straight  sides  forming 
the  angular  groove  are  connected  to  the  rib  between  the 
grooves  by  curves  as  in  Fig.  68. 

Another  form  which  has  much  in  its  favor,  especially 
for  the  independent  rope  system,  is  the  circular  arc  groove, 
in  which  the  sides  are  formed  by  arcs  of  circles, 


188 


Working  proportions  for  this  groove  are  given  in  Fig. 
69,  in  which  the  unit  d  equals  the  diameter  of  rope.     It 


FIG.  67. — RIM  SECTION — ENGLISH  FORM 


FIG.  68. — RIM  SECTION— IMPROVED  ENGLISH  FORM. 
will  be  noticed  that  the  centre  for  the  curve  is  located  at 
the  intersection  of  a  line  drawn  through  the  centre  of  the 


ROPE-DRIVING. 


189 


rope  at  an  angle  of  22J°  with  the  horizontal  and  a  line 
drawn  through  the  tops  of  the  dividing  ribs;  the  angle  of 
the  groove  embraced  by  the  rope  is  thus  dependent  upon 
the  position  of  the  latter  in  the  groove :  in  its  normal  posi- 
tion the  angle  is  one  of  45°.  Tho  Walker  groove,  as  now 
used  in  rope-pulleys  made  by  Eraser  &  Chalmers  of  Chicago 
and  the  Walker  Mfg.  Co.  of  Cleveland,  has  its  sides  formed 
with  circular  arcs  similar  to  Fig.  69,  but  the  angle  of  the 


'Elec.  World 
I  UNIT=d 

ANGLE  OF  GROOVE  45°WHEN  ROPE  IS  IN  ITS  NORMAL  POSITION 

FIG.  69. — RIM  SECTION — GROOVES  WITH  CIRCULAR  ARCS. 

groove  is  more  acute;  at  the  point  of  tan  gen  cy  when  the 
rope  simply  rests  in  the  groove  the  angle  measures  33°. 

Rope-pulleys  for  ordinary  transmissions  are  usually  flat 
on  the  inside  of  the  rim,  or  slightly  tapered  as  shown  by 
the  full  lines  in  Fig.  69.  In  large  wheels,  however,  the 
rim  is  frequently  swept  up,  and  where  the  weight  with  a 
flat  rim  would  be  greater  than  required  for  strength  or 
steady  running,  the  inside  is  hollowed  out  as  shown  by 
dotted  line.  This  gives  a  more  nearly  uniform  section 
and  makes  a  stronger  and  lighter  wheel,  but  the  expense  of 
construction  is  greater.  Guide-pulleys  or  idlers  should  be 
made  with  a  semicircular  groove,  so  that  the  rope  runs 
upon  the  bottom  instead  of  being  wedged  between  the 
sides,  as  in  grip-pulleys.  Some  engineers  maintain  that 


190 


HOPE-DRIVING. 


the  wedge  groove  should  be  used  in  all  cases,  and  many 
plants  are  in  operation  having  V-shaped  grooves  in  the 
idlers;  but  we  believe  the  most  satisfactory  results  will  be 
obtained  by  using  a  groove  in  which  the  rope  runs  on  the 
bottom.  Of  these  there  are  two  general  forms — the  one 
in  which  the  rope  has  considerable  play  in  the  groove,  and 
the  other  of  such  form  that  the  rope  is  embraced  by  a 
portion  of  the  surface  of  the  groove:  the  first  is  used  more 
particularly  for  single  ropes,  while  the  latter  is  adapted  to 
any  number  of  wraps. 

A  modification  of  the  first  form  is  frequently  used  for  a 
number  of  routes  as  shown  in  Fig.  70. 


FIG.  70.— FORM  OF  GROOVE  FOR  GUIDE  PULLEYS. 

The  proportions  shown  in  Fig.  71  will  be  found  very 
satisfactory  for  single  or  multiple  groove  guide-pulleys,  in 
which  the  pitch  is  equal  to  that  used  with  the  grip-pulleys 
represented  in  Fig.  69;  the  general  design  conforms  closely 
to  the  latter,  brt  the  pulley  is  somewhat  lighter. 

For  shaft-pulleys  and  the  smaller  sizes  of  fly-wheels  not 
exceeding  about  nine  feet  in  diameter  the  casting  is  usually 
made  in  one  piece,  unless  a  "split"  pulley  is  required.  In 
order  to  relieve  the  wheel  in  cooling  the  arms  were  formerly 


ROPE-DRIVING. 


191 


given  a  curved  or  S  shape,  as  this  form  yields  more  readily 
and  is  supposed  to  conform  to  the  unequal  contraction  of 
the  rim  and  hub  of  the  pulley  in  cooling.  With  properly 
proportioned  wheels,  however,  and  with  due  care  in  the 
foundry,  straight- armed  pulleys  may  be  cast  as  strong  as 
those  of  curved  outline,  and  as  the  former  are  lighter,  neater, 
and  cheaper  than  the  latter  they  are  now  almost  universally 
employed.  In  the  larger  single-casting  pulleys  the  hubs 
are  frequently  split  in  order  to  favor  the  arms  in  cooling: 


FIG.  71. — FORM  OF  GROOVE  FOR  GUIDE  PULLEYS. 

when  split  by  a  diametral  plane  the  two  portions  at  the 
hub  are  generally  secured  by  bolts  and  nuts  as  in  Fig.  72, 
or  less  frequently  by  pins  and  cotters;  when  split  in  three, 
wrought-iron  or  steel  rings  are  shrunk  onto  the  ends  of  the 
hub,  which  is  turned  to  receive  them,  as  represented  in  Fig. 
73.  Occasionally  rings  are  shrunk  on  and  bolts  used  as 
well,  but  this  precaution  is  not  common  practice. 

There  is  no  general  rule  by  which  the  number  of  arms 
maybe  determined.  For  small  rope-pulleys  the  number  is 
usually  six,  while  for  the  larger  sizes  six  or  eight  arms  are 
used  for  pulleys  cast  in  one  or  two  parts ;  but  these,  in  some 


192 


ROPE-DRIVING. 


cases,  have  two  sets  of  arms  as  shown  in  Fig.  74,  which 
represents  a  78-inch  pulley  made  by  the  Robert  Poole  & 
Son  Co.  for  the  Providence  Cable  Tramway  Co.  The  usual 


FIG.  72. 


FIG.  73. 


run  of  wheels  from  9  feet  to  15  feet  in  diameter  are  cast 
in  halves,  six  or  eight  arms  being  employed. 

Occasionally,  however,  narrow  rope  fly-wheels,  of  diam- 
eters up  to  18  feet,  are  made  in  two  pieces  only,  with  eight 
arms. 


ROPE-DKIVING. 


193 


Moderate-sized  wheels  arc  frequently  cast  in  one,  but 
arranged  so  as  to  be  readily  split  in  two  afterwards. 


Cottoh  Ropes 


Pulleys  cast  in  halves  having  light  rims  should  have 
double  arms  along  the  line  of  separation,  as  shown  in  Fig. 
75,  which  represents  a  form  of  split  pulley  made  by  Fair- 


HOPE-DRIVING. 


ROPE-DRIVING.  195 

banks,  Morse  &  Co. ;_  this  prevents  undue  bending  action, 
and  is  otherwise  the  better  able  to  resist  the  effect  of  cen- 
trifugal force  due  to  the  added  weight  at  the  point  of 
connection. 

For  diameters  ranging  from  15  feet  to  20  feet  the  usual 
practice  is  to  make  the  pulley  in  six  or  eight  segments  with 
as  many  arms,  although  some  makers  prefer  ten  segments; 
from  20  to  26  feet  ten  segments  are  usually  adopted.  For 
wheels  from  about  26  to  32  feet  twelve  segments  and  as 
many  arms  are  generally  used,  while  larger-sized  wheels  are 
built  up  of  fourteen,  sixteen,  and  even  more  segments — 
sometimes  as  many  as  twenty-four  being  employed.* 

When  rope-pulleys  are  made  in  halves  or  are  built  up  it 
is  important  that  the  connecting  bolts  and  flanges  in  the 
rim  should  be  strong  enough  to  resist  the  maximum  stresses 
that  may  occur  in  the  joint. 

Ordinarily  in  rope-pulleys  the  net  section  of  bolt  area  is 
about  12  per  cent  of  the  section  at  the  joint,  but  this  ranges 
from  25  per  cent  to  about  6  per  cent  in  different  cases. 
Various  shop  rules  are  used  by  designers  for  obtaining  the 
bolt  section,  many  of  which  give  the  bolt  area  directly  in 
terms  of  the  cross-section  of  the  rim.  While  such  formulae 
mny  give  satisfactory  results  for  an  assumed  average  rim 
speed,  it  is  better  to  design  the  joint  in  large  wheels  from 
a  consideration  of  the  particular  conditions  in  each  case. 

In  very  slow-moving  heavy  wheels  the  principal  stress 
may  be  that  due  to  the  weight  of  the  wheel  itself,  but  in 
rope-pulleys  and  fly-wheels  the  rim  has  usually  a  high 
velocity,  and  the  strain  due  to  centrifugal  force  in  the  rim 
is  the  principal  factor  in  determining  the  bolt  section;  it  is 
obvious,  however,  that  in  high-speed  heavy  wheels  both 
influences  should  be  taken  into  consideration. 

The  tension  in  the  rim  produced  by  centrifugal  force  is 


*  American  Machinist,  Feb.  16,  1893. 


196 


KOPE-DK1V1NG. 


equal  to  one  half  the  centrifugal  force  due  to  the  weight 
and  velocity  of  the  rim  multiplied  by  the  ratio  of  diameter 
to  semi-circumference,  or  equal  to 

F       2r       F 

^-2  V  __  —  £» 
c\      ^   -  ~~  --  • 

2        nr       n 

As  this  force  is  resisted  at  each  end  of  a  diameter,  the 
strain  Th,  or  hoop-tension,  acting  at  either  end  will  be  one 
half  the  above;  hence 


in  which  F0  has  the  usual  value  0.00034  WRN*, 
where  W  =  entire  weight  of  rim  ; 

R  =  effective  radius  of  rim  in  feet; 
N  =  revolutions  of  wheel  per  minute. 
The  tension  at  each  end  of  a  diameter  due  to  the  weight 


of  the  rim  is  evidently  equal  to 


1fW\ 

-  -- 


iS\ 

in  the  rim  due  to  F  and  W  will  be 


therefore  the  stress 


If  the  bolts  could  be  placed  at  the  point 
of  application  of  this  force,  their  effective 

o 

section  A  would  be  -,  where  /is  the  allow- 

able stress  in  the  bolts. 

In  the  actual  construction  the  centre  of 
bolts  must  be  placed  at  some  distance 
from  the  point  of  application,  in  which 
case  the  bolts  are  subjected  to  an  addi- 
tional strain  due  to  the  bending  moment 
at  the  joint,  as  shown  in  Fig.  76.  If  this  bending 
moment  is  taken  up  by  the  rim,  the  bolt  section  will  be 


ROPE-DRIVIKG.  197 

^as  before;  but  if  the  rim  and  bolt  flanges  are  rigid  and 

resist    any    deformation,    the    total    strain    comes    upon 
the   bolts.     Under   these   conditions   the   net  section    A 

<y 

may   be  obtained  by   multiplying  j   by   the  ratio  of  the 

leverage  y  of  the  force  8  to  the  leverage  x  of  the  resist- 
ance in  the  bolt;  hence  if  we  assume  that  x  =  \y,  a  com- 

mon value,  we  shall  obtain  A  =  2-.. 

By  using  studs  with  nuts  at  each  end  the  bolt  may  be 
placed  nearer  the  rim,  in  which  case,  for  the  same  depth  of 

flange,  the  ratio  -  may  be  made  less  than  2.     In  order  to 

obtain  ample  strength  at  the  rim,  the  bolt  section  should 
be  determined  on  the  supposition  that  the  full  bending 
moment  will  be  thrown  upon  the  bolts.  In  this  case  the 
bolt-flanges  or  lugs  must  be  able  to  resist  any  bending  due 
to  the  force  S.  The  bolt-flanges  will  be  as  strong  as  the 
rim  if  we  make  the  respective  moments  of  resistance  equal 
to  each  other;  hence 
if  b  =  face  of  wheel; 

b'  —  breadth  of  bolt  -flange   minus  the  width  of  bolt- 
holes; 

t  —  mean  thickness  of  rim; 

t'  —  thickness  of  bolt-flange  or  lug, 
then 


or        =         ,. 

This  thickness  t'  may  be  reduced  somewhat  if  strength- 
ening ribs  are  carried  from  the  rim  to  the  lower  edge  of 
the  bolt-flange,  as  shown  in  Fig.  76.  In  the  above,  t 
was  taken  as  the  mean  thickness  of  rim  reduced  to  a  rec- 
tangle, and  no  account  has  been  taken  of  the  strength- 
ening ribs  and  flanges.  If  the  leverage  of  the  bolts  is 


198  ROPE-DRIVING. 

assumed  as  one  half,  we  may  obtain  the  total  net  section 
of  the  bolts  as  follows : 


Since  F0  =  0.  00034  WRN\ 

.OOOlOS  WRN*      ~ 


=  -ifo. 


In  the  above  formula  /  may  be  taken  equal  to  6000 
pounds,  for  bolts  up  to  1£  or  1-J-  inches  diameter,  but  for 
larger  sizes  8000  to  9000  pounds  may  be  used. 

If  t  =  the  mean  thickness  and  b  —  breadth  of  rim  in 
inches,  the  weight  W  may  be  determined  from 


W=%7rJtx  1Mb  X  0.26 
=  IQ.GRtb. 

If  the  rim  is  properly  bolted,  the  principal  strains  to 
which  the  hub-bolts  are  subjected  are  those  due  to  the 
weight  of  hub  and  arms  and  the  tension  produced  by  key- 
ing to  the  shaft. 

As  the  weight  of  hub  and  arms  in  large  pulleys  is  usu- 
ally much  less  than  the  weight  of  rim,  it  will  be  seen  that 
the  strain  on  the  hub-bolts  due  to  the  weight  and  centrif- 
ugal force  of  these  parts  will  be  less  than  that  on  the  rim- 
bolts. 

In  practice  some  makers  design  the  hub-joint  so  that  the 
net  section  of  all  the  bolts  is  equal  to  the  bolt  section  in 
one  edge  of  the  rim-joint;  this,  however,  is  not  usual 
practice.  An  inspection  of  a  great  many  pulleys  made  by 
various  manufacturers  shows  that  the  hub-bolt  section  is 
often  twice  as  great  as  the  rim-bolt  section;  but  in  many 
of  these  cases  the  rim-bolting  is  very  weak. 

It  is  safe  to  make  the  bolt  area  the  same  in  each  case; 
but  if  the  rim-joint  is  made  according  to  the  method 


HOPE-DRIVING. 


199 


indicated,  the  bolts  in  the  hub  will  usually  be  sufficiently 
strong  if  their  total  effective  section  is  equal  to  that  in 
one  rim  joint:  for  light  pulleys,  however,  this  should  be 
increased.* 

The  section  of  arm  for  those  pulleys  in  which  the  arms 
and  hub  are  cast  together  is  usually  elliptical  or  segmental, 
as  shown  in  Fig.  77. 


FIG.  77.— SECTIONS  OF  ARMS. 

The  elliptical  form  as  here  given  is  so  proportioned  that 
its  minor  axis  is  one  half  its  major;  in  the  segmental  form 
the  minor  axis  is  four  tenths  the  major. 

The  cross-section  of  the  arm  may  be  determined  by  con- 
sidering it  as  a  beam  fixed  at  one  end  and  loaded  at  the 
other  with  the  force  P  due  to  the  pull  of  the  ropes.  The 
bending  moment  on  each  arm  will  then  be 

Pr 
iw  — 

b  ~  ~N~' 

-LV  a 

assuming  that  the  load  is  divided  equally  among  the  num- 
ber of  arms,  Na. 

*  For  complete  analysis  of  strains  in  fly-wheel  rims,  see  Prof.  Un- 
win's  discussion  in  his  "Machine  Design,"  vol.  ir.  Also.  Mr. 
Stanwood's  paper  in  vol.  xiv.  Trans.  A  S.  M.  E.,  with  Prof. 
Lanza's  discussion. 


200  ROPE-DRIVIKG. 

Since  the  bending  moment  is  equal  to  the  modulus  of 
the  section,  Z,  multiplied  by  the  stress,/,  in  the  material, 
we  have 

Jft=fZ; 

For  an  elliptical  section  whose  major  axis  is  h  and  minor 
axis  OAh, 

Z=^XM; 
hence 


If  we  let  Tlf  the  maximum  tension  in  the  rope,  equal 

T 

200d2  pounds,  and  -~  =  2,  we  shall  have  the  load  acting  on 

•*» 
each  arm 

P  =  lOOd'n, 

where  n  is  the  number  and   d  the  diameter  of  ropes. 
Therefore  by  substituting  this  value  of  P  in  equation  (18) 


there  is  obtained  h  = 


0.04/2VV 


Assuming  that/=  2500  pounds  per  square  inch  for  cast 
iron  and  that  Na  equals  6,  we  have 


h  =  0.44 

where  D  =  pitch  diameter  of  pulley  and  li  is  the  major 
axis  of  arm  produced  to  centre  of  wheel,  both  in  inches.  As 
the  smaller  pulleys  require  a  larger  margin  of  strength,  on 
account  of  their  greater  liability  to  breakage  in  handling 
and  casting,  the  above  formula  should  be  modified  by  intro- 
ducing a  constant;  if  we  let  this  constant  equal  -J  inch,  a 
suitable  value  for  the  width  of  arm  will  be  obtained  from 


.5"  .....     (19) 


HOPE-DRIVING.  201 

As  the  centrifugal  force  set  up  in  the  rim  of  the  pulley 
causes  an  additional  stress  in  the  metal,  the  value  of  / 
should  be  chosen  with  reference  to  the  speed  at  which  the 
pulley  is  to  run.  The  higher  speed  not  only  induces  a 
greater  stress  in  the  material,  but  the  liability  of  failure 
due  to  vibration  or  shocks  is  greatly  increased.  The  stress 
in  the  rim  due  to  centrifugal  force  may  be  obtained  by 
considering  the  pulley  as  a  cylinder  subjected  to  a  force 
p(  =  F0)  per  square  inch.  The  thickness  of  a  thin  cylinder 
to  resist  rupture  may  be  obtained  from 

._pD"_pr 

~w~  r 

in  which  t  =  thickness  in  inches; 

p  =  pressure  per  square  inch; 
r  =  radius  of  cylinder  in  inches; 
f  =  allowable  stress  in  pounds. 

In  this  case 


W=  weight  in  pounds  =  0.261  pounds  per  cubic  inch; 
v  =  velocity  of  rim  in  feet  per  second  ; 
g  =  32.16; 

T 

R  =  radius  of  rim  in  feet  =  —  ; 

1/w 

hence 

Fr       W       r        Wtfr 


and 

12TFV 

7  ^    ' 


202 


If  FQ  act  on  one  square  inch  of  pulley  whose  thickness 
is  unity,  we  shall  have 


< 
=  ~,  very  nearly; 

this  is  the  stress  in  rim  per  square  inch  of  section  due 
to  centrifugal  force  alone. 


When  v  in  feet  per  second            = 

50 

60 

70 

80 

90 

100 

150 

200 

"      F"     "      "    minute 

3000 

8(500 

4200 

4800 

5400 

6000    9000 

12000 

Stress  /'  in  pounds  due  to  centrif- 

ugal tension 

250 

360 

490 

640 

810 

1000    2250 

4000 

The  stresses  due  to  the  pull  of  the  ropes,  and  those  due 
to  contraction  in  cooling,  are  additional  to  those  here  given, 
hence  /  should  be  taken  sufficiently  low  to  allow  for  the 
various  stresses  which  may  be  set  up  in  the  pulley.  If  we 
assume  that  the  working  stress  should  not  ordinarily  exceed 
3500  pounds  per  square  inch  for  cast-iron  pulley  arms,  and 
that  it  should  be  less  for  high  speeds  where  the  dynamic 
effect  of  shock  and  vibration  is  greater,  a  suitable  value 
for  cast  iron  may  be  obtained  from  the  empirical  formula 

50000 


5+   fF 

From  this  formula  the  annexed  values  of  /  have  been 
calculated;  it  will  be  noticed  that  the  value  of  /  used  in 
formula  (19),  namely,  2500  pounds  per  square  inch,  corre- 
sponds to  a  velocity  of  rim  equal  to  about  3500  feet  per 
minute. 


Velocity  of  rope  in  feet  per  minute  F  = 
Allowable  stress                                   /  = 

1200 
9-300 

1800 
2900 

2400 
2700 

3600 
2450 

4800 
2300 

0000 
2150 

ROPE-DRIVING.  £03 

The  arms  should  taper  toward  the  rim  TV  inch  per  inch 
of  length,  that  is,  -js  on  each  side,  but  in  no  case  should 
the  width  of  arm  at  the  rim  be  less  than  two  thirds  its 
width  at  the  centre  of  shaft. 

Very  wide  pulleys  in  which  the  proportions  for  single 
arms  would  be  inconveniently  large  may  be  made  with  two 
or  three  sets  of  arms;  in  such  cases  they  may  be  considered 
as  two  or  three  separate  pulleys  combined  in  one,  except 
that  the  proportions  of  the  arms  should  be  0.8  to  0.7  times 
that  of  single  -arm  pulleys.* 

In  designing  the  hubs  of  wheels  practice  varies  consider- 
ably. Some  authorities  give  the  thickness  of  metal  around 
the  eye  in  terms  of  the  pulley  diameter  only,  others  take 
into  account  the  diameter  of  pulley  and  also  the  diameter 
of  shaft  or  the  breadth  of  face,  while  the  length  is  variously 
given  in  terms  of  the  diameter  of  shaft  or  the  face  of 
pulley,  or  both. 

For  rope-pulleys  if  the  thickness  of  metal  in  the  hub  is 
made  proportional  to  the  diameter  of  pulley  and  also  to  the 
diameter  of  shaft,  and  if  the  length  of  hub  is  made  to  vary 
with  the  number  and  size  of  ropes  and  the  diameter  of 
shaft,  we  believe  the  requirements  of  strength  and  good 
proportions  will  be  best  attained. 
If  D  —  diameter  of  pulley, 
ds  =         "         "  shaft, 
dh  =         "        "  hub, 
d  =        "        "  rope, 
n  —  number  of  ropes, 
Lh  —  length  of  hub, 

then  the  diameter  and  length  of  hub  may  be  obtained 
from  the  following  formulae,  which  have  been  deduced  from 
the  proportions  of  a  large  number  of  rope  pulleys  made  by 
representative  manufacturers  : 


*  Reuleaux,  "  Constructeur." 


204 


ROPE-DKIVIHG. 


and 


Lh  =  0.6^1 


For  loose  or  idle  pulleys  the  diameter  of  hub  may  be 
made  less  than  that  given  by  the  above  formula,  which 
allows  for  keying.  In  general  the  thickness  of  metal 
around  the  eye  in  an  idle  pulley  may  be  taken  as  about 
two  thirds  as  great  as  that  in  fixed  pulleys  of  the  same 
diameter  and  face.  The  length  of  hub  in  idlers  should  be 
sufficient  to  give  a  good  bearing  surface,  and  may  vary 
from  two  to  three  times  the  diameter  of  shaft — depending 
somewhat  upon  the  speed  of  rotation. 

n 


FIG.  78,— METHOD  or  JOINING  ARMS  AND  RIM. 

In  the  construction  of  large  rope-pulleys  which  are 
made  in  segments,  the  usual  method  is  to  bolt  the  rim  sec- 
tion to  the  arms  at  the  ends  of  the  segment,  as  shown  in 
Pig.  78. 

When  the  rim  segments  are  joined  midway  between  the 
arms  as  in  Fig.  79  the  several  connections  are  simplified  to 
some  extent,  but  with  this  method  of  connection  there  is  a 
decided  tendency  for  the  joint  to  open  under  the  influence 
of  centrifugal  force,  which  is  increased  somewhat  by  the 
weight  of  the  connecting  flanges  and  bolts.  With  the  rim- 
joints  at  the  junction  with  the  arms,  however,  a  rigid  con- 


ROPE-  DRIVING. 


205 


nection  between  adjacent  arms  is  obtained,  and  the  centrif- 
ugal effect  of  the  rim  tends  to  increase  the  tension  in  the 
arm  without  opening  the  joint. 

Examples  of  both  methods  of  connection  are  given  in 
Figs.  80  to  92,  which  also  represent  some  of  the  details  of 
construction  in  various  built-up  rope-pulleys. 

As  will  be  noticed,  these  built-up  wheels  are  made  with 
a  large  central  boss  or  hub,  usually  in  one  piece,  but  some- 
times in  two,  provided  with  seats  for  the  arms,  or  bored, 


FIGj  79. — METHOD  OF  JOINING  ARMS  AND  RIM. 

either  tapering  or  straight,  to  receive  the  ends  of  the  arms, 
which  are  turned  to  fit  the  holes. 

The  arms  are  usually  of  round  or  elliptical  section,  cast 
hollow;  but  other  forms  are  common.  Thus  in  Figs.  85 
and  90  the  arm  is  of  cruciform  section,  while  in  Fig.  81  a 
modification  of  the  H  section  is  used. 

In  Fig.  80  we  have  an  interesting  form  of  wheel  patented 
some  years  ago  by  Mr.  James  Barbour,  of  Combe,  Barbour 
&  Combe,  Belfast.*  The  rim  of  the  wheel  is  made  in  seg- 

*  Engineering,  Sept.  7,  1888.; 


206 


ROPE-DRIVING. 


nients,  and  is  attached  to  the  boss  or  hub  by  means  of  the 
bolts  or  tie-rods  d,  extending  from  the  rim  to  the  boss  as 
shown,  and  passing  through  the  arms,  which  are  made  hol- 
low for  the  purpose.  The  rim  and  hub  are  recessed  to  re- 
ceive projecting  pieces  a' a'  on  the  ends  of  the  arms.  The 


FIG.  80. — BUILT-UP  WHEEL.     (Barbour.) 

head  of  the  tie-rod  d'  can  be  made  to  form  part  of  the 
periphery  of  the  wheel.  The  tie-rods  have  slots  formed  in 
their  inner  ends  to  receive  keys  bp,  which  keys  are  passed 
into  the  hub  through  holes  parallel  to  the  axis  of  the  shaft, 
and  in  this  way  the  rim  of  the  wheel  is  secured  to  the  hub. 


FIG.  81.— THIRTY-FOOT  ROPE-WHEEL.     (Hick.) 


FIG.  81  (a). 
SECTION  OF  GROOVE. 


FIG.  81  (5)  and  (c). 
SECTION  AT  Boss. 


208  ROPE-DRIVING. 

The  object  of  the  arrangement  is  to  obviate  the  use  of 
a  number  of  bolts  for  connecting  the  rim  to  the  arms; 
moreover,  the  tie-rods  withstand  the  centrifugal  force  of 
the  rim,  and  the  wheel  can  be  driven  with  safety  at  a 
high  rate  of  speed.  The  usual  projections  from  the 
principal  surfaces  are  also  eliminated,  and  the  currents 
of  air  generated  by  the  high  velocity  of  the  rim  are  less- 
ened. This  latter  feature  is  one  the  value  of  which  is  be- 
coming more  recognized  lately,  especially  in  rope-driving, 
when  a  high  velocity  of  rim  is  especially  advantageous. 

To  lessen  the  fan  action  set  up  in  a  large  fly-wheel,  we 
frequently  find  the  arms  boarded  up,  provision  being  made 
for  this  in  the  wheel. 

Fig.  81  represents  a  fly-wheel  30  feet  in  diameter  made 
by  Hick,  Hargreaves  &  Co.,  Bolton,  for  a  1000  horse-power 
condensing-engine.*  The  weight  of  the  wheel  is  54  tons. 
The  rim  is  6  feet  wide,  and  is  grooved  for  27  cotton  ropes 
5  inches  in  circumference,  or  nearly  If  inches  in  diameter, 
at  2-J  inches  pitch.  The  grooves  are  shown  in  detail 
at  (a). 

The  rim  is  constructed  of  twelve  segments  with  twelve 
arms.  The  segments  are  planed  at  the  joints,  and  are 
bolted  together  with  eight  bolts  and  nuts,  of  which  the 
two  next  the  arms  are  If  inches  in  diameter  and  the  others 
are  1£  inches.  Each  arm  is  secured  with  four  2-inch  T- 
head  bolts  and  nuts  to  the  rim  segments,  two  to  each  of 
two  segments.  The  boss,  nave,  or  centre,  shown  in  sec- 
tion at  (£),  is  6  feet  i  inch  in  diameter,  or  7  feet  across  the 
platforms,  or  slightly  raised  plane  surfaces,  on  which  the 
arms  take  their  bearings.  It  is  18  inches  wide  at  the  rim 
and  2  feet  wide  at  the  bearing  on  the  shaft.  Twelve  sock- 
ets are  bored  out  to  receive  the  ends  of  the  arms.  The 
arms  are  formed  approximately  of  H  section,  as  at  (c),  and 

*  D.  K.  Clarke,  "  Steam-Engine,"  vol.  in. 


ROPE-DRIVING.  209 

measure  14  inches  by  9  inches  at  the  centre  and  9-J  inches 
by  6|  inches  at  the  rim.    They  are  turned  conically  to  fit 


FIG.  82.— TWENTY-FOUR-FOOT  ROPE- WHEEL.    (Walker.) 

the  holes  in  the  centre,  tapering  from  12  inches  in  diam- 
eter at  the  outer  ends  of  the  sockets  to  7-J  inches  at  the 


210  HOPE-DRIVING. 

inner  ends  in  a  total  length  of  2  feet  5  inches.  The  arm 
is  keyed  into  the  centre  with  two  cotters,  each  24  inches 
long,  1  inch  thick,  tapering  in  width  from  3f  inches  to  3^ 
inches,  or  -J  inch  in  24  inches.  They  are  driven  in  one 
from  each  side  of  the  nave  reversely,  and  make  up  a 
united  width  of  6f  inches.  To  join  the  rim  the  arm  is  ex- 
panded into  a  flat  flange  21  inches  by  18  inches  and  4 
inches  thick,  through  which  the  four  holts  already  named 
are  passed.  The  opening  at  the  centre  is  25  inches  in  di- 
ameter, or  2  inches  larger  than  the  shaft.  The  wheel  is 
fixed  on  the  shaft  with  six  keys  5  inches  wide  and  If 
inches  thick,  bearing  on  six  flat  seats  formed  on  the  shaft, 
with  a  taper  of  £  inch  to  the  foot. 

Pig.  82  represents  a  somewhat  similar  method  of  con- 
nection employed  by  the  Walker  Manufacturing  Company, 
Cleveland.  In  this  pulley  the  arms  are  of  round  or  pipe 
section,  and  present  a  much  neater  appearance  and  better 
proportions  than  obtain  in  the  previous  pulley. 

This  wheel  was  made  for  the  Baltimore  City  Passenger 
Railway  Station,  and  is  24  feet  in  diameter,  30  inches  face, 
grooved  for  ten  2 -inch  ropes  at  2J  inches  pitch.  The  de- 
tails are  shown  in  Figs.  83  and  84. 

The  centre  is  a  single  casting  5  feet  6  inches  in  diam- 
eter, bored  to  receive  the  arms,  of  which  there  are  ten. 
Each  arm  is  10J  inches  in  diameter  near  the  hub  and  8 
inches  in  diameter  near  the  rim,  where  it  is  expanded  into 
a  flange  and  bolted  to  the  rim  segments.  The  bolts  con- 
necting the  arms  to  the  segments  and  the  segments  to 
each  other  are  all  turned  1||  inches  in  diameter,  and  fit  in 
drilled  and  reamed  holes. 

The  shoulder  on  the  arms  at  the  hub  is  15  inches  in  di- 
ameter, and  is  faced  to  fit  the  machined  seats  on  the  boss. 
By  this  means  perfect  alignment  is  obtained,  and  no 
motion  of  the  parts  is  possible  unless  the  key  shears. 

Fig.  85  represents  a  rope-pulley  designed  by  Mr.  F.  Van 


KOPE-DEIVIKG. 


211 


Bolt  holes  drilled  and  reamed 
Bolts  turned 


.  83,— DETAILS  OF  PULLEY  SHOWN  IN  FIG.  83.  SECTION  OF  ARM, 


212 


HOPE-DRIVING. 


DETAIL  OE  ARM  AT  CENTER 


FIG.  84,— DETAILS  or  PULLEY  SITOWN  IF  FIG.  83, 


214 


Vleck  for  the  San  Diego  Cable-railroad  Power-station.*  It 
is  25  feet  in  diameter,  42|  inches  face,  grooved  for  twelve 
2-inch  cotton  ropes  at  3^  inches  pitch.  This  drum  is  com- 
posed of  ten  separate  segments  and  ten  arms,  bolted  and 
keyed  midway  in  each  segment.  The  central  hub  is  4£ 


feet  in  diameter  across  flats,  and  is  proportioned,  as  in  fact 
the  whole  wheel  is,  with  special  regard  to  lightness. 

The  designs  shown  in  Figs.  86  to  89  are  for  the  large 
rope-pulleys  used  in  the  Fifty-first  Street  and  Houston  Street 

*  Trans.  A.  S.  M.  E.,  vol.  xn.  p.  77. 


j?IG   87 —THIRTY-TWO-FOOT  ROPE- WHEEL.    (Walker  Mfg.  Co.) 


216 


ROPE-DKIVLtfG. 


SECTION  ON  LINE 
A   B 


SECTION  O;N  LINEJ 

C   D 


HALF  SECTION 
ON  LINE  E   F 

FIG.  88.— THIRTY-TWO- FOOT  ROPE-WHEEL.    (Walker  Mfg.  Co.) 


ROPE-DRIVIKG. 


stations  of  the  Broadway  Cable  Road.  The  first  wheel, 
Fig.  86,  is  32  feet  in  diameter,  37^  inch  face,  and  is  grooved 
for  13  cotton  (Lambeth)  ropes  2  inches  in  diameter,  2f 
inches  pitch.  The  speed  of  the  ropes  is  very  low — only 
1877  feet  per  minute.  Mr.  M.  W.  Sewall  *  gives  the  fol- 
lowing particulars  regarding  the  details  of  this  wheel,  as 
at  first  designed.  It  will  be  seen  by  reference  to  the  figure 
that  the  arms  of  the  drum  are  secured  to  the  centre  by 
clamping  two  turned  portions  on  the  inner  end  of  each 
arm,  between  heavy  cast-iron  disks,  one  of  which  is  cast 


FIG.  89.— SECTION  OF  RIM  OP  PULLEY 
SHOWN  IN  FIGS.  87  AND  88. 


as  a  flange  on  the  hub  and  the  other  acts  as  a  fol- 
lower. These  are  bolted  up  a  slight  distance  apart,  and 
the  holes  for  the  arms  are  bored  accurately  to  fit  the  turned 
portions  of  the  same.  The  bolts  are  then  loosened,  the 
arms  put  in  place,  and  the  follower  bolted  hard  up  to  the 
hub  flange.  A  taper  pin  is  then  driven  into  a  reamed  hole 
through  each  arm  and  the  parts  of  the  centre,  and  held  in 
place  by  a  nut.  The  arms  and  the  centre  are  then  practi- 

*  For  complete  description  of  the  cable-driving  machinery  see 
American  Machinist,  May  24  and  31,  1894. 


HOPE-DRIVIKG. 


frothy  $  Poales, 

FIG.  90.— HEAVY  ROPE-WFIEEL,  TWENTY-SEVEN  FEET  DIAMETER. 
(Wetherill  &  Co.) 


819 


FIG.  91.— SECTION  OF  TWENTY-SEVEN-FOOT  WHEEL, 


220 


cally  one  piece,  and  no  working  of  the  parts  among  them- 
selves is  possible.  None  of  the  bolts  secure  an  individual 
arm,  and  no  stress  transmitted  to  the  centre  through  the 
arms  brings  any  of  them  into  direct  tension  ;  several  bolts 
might  be  broken  without  detriment  to  the  structure.  This 
drum  would  be  improved,  so  far  as  convenience  of  manu- 
facture is  concerned,  by  a  shoulder  on  the  arm  to  deter- 
mine its  distance  from  the  centre  when  assembling  in  the 
shop.  The  arms  are  attached  to  the  rim  in  the  usual  man- 
ner, but  are  rather  light  in  proportion  to  the  rest  of  the 
design.  These  pulleys  were  redesigned  to  suit  the  contrac- 
tor for  the  work,  The  Walker  Manufacturing  Co.  of  Cleve- 
land, and  were  constructed  in  a  manner  similar  to  that 
shown  in  Fig.  87,  except  that  single  arms  and  centre  were 
used. 


Bradlev  <f  Poates.  Ennr's.  N.  Y. 


FIG.  92. — SECTION  OF  GROOVES  FOR  WHEEL  SHOWN  IN 
FIGS.  90  AND  91. 

Figs.  87,  88,  and  89  represent  the  32-foot  drums  used  in 
the  Houston  Street  station,  and  are  much  wider  and 
heavier  than  those  used  in  the  Fifty-first  Street  station. 
As  will  be  noticed,  these  drums  are  8  feet  4  inches  face, 
grooved  for  thirty-four  2-inch  ropes  at  2J  inches  pitch,  and 
arranged  with  two  sets  of  arms  and  two  centres.  The  weight 
of  each  puHey  is  104  tons. 

As  constructed,  the  arms  have  heavy  flanges  bolted  to  the 
centre,  into  which  a  turned  projection  on  the  end  of  the 


KOPE-DRIVItfG.  221 

arm  is  accurately  fitted.  In  addition  to  this  two  of  the  four 
bolts  securing  each  arm  to  the  centre  is  turned  to  fit  reamed 
holes  in  the  boss ;  these  bolts  thus  act  as  dowels  and  pre- 
vent any  working  of  the  parts.  The  bolts  are  calculated 
to  resist  the  maximum  tension  in  the  arms,  and  are  2T\  and 
2£  inches  in  diameter  ;  in  the  same  way  the  arms  are  bolted 
to  the  rim,  two  of  the  four  bolts  fitting  in  reamed  holes. 
The  arms  are  of  hollow  elliptical  section  15  by  10  inches  at 
the  hub,  and  10  X  7-J  inches  at  the  rim.  Each  centre  is  8 
feet  diameter  across  flats ;  they  are  bored  26  inches  in  di- 
ameter, and  are  3  feet  long  in  the  bore. 

Large  rope-pulleys  from  20  to  32  feet  in  diameter  are, 
when  extra  wide,  frequently  made  with  two  centres  as  well 
as  two  sets  of  arms,  as  just  noted  in  Fig.  87.  This  is  also 
seen  in  Figs.  90  to  92,  which  represent  a  rope-driving  wheel 
26  feet  9  inches  pitch  diameter,  8  feet  face,  grooved  for 
twenty-four  3-inch  ropes,  at  3J  inches  pitch.  This  pulley 
was  designed  and  built  by  Robt.  Wetherill  &  Co.  of  Ches- 
ter, Pa.,  and  consists  essentially  of  two  separate  and  inde- 
pendent drums,  flanged  and  bolted  together  at  the  rim. 
Each  centre  is  made  of  two  separate  disks  6  feet  6  inches 
in  diameter,  bored  and  faced  on  the  inside.  The  arms,  of 
which  there  are  twelve,  are  of  cruciform  section  between 
the  boss  and  the  rim,  where  they  are  flanged  and  bolted  to 
the  rim  segments  in  the  usual  manner.  At  the  centre  the 
arms  are  wedge-shaped,  8  inches  thick,  and  are  so  propor- 
tioned that  when  accurately  planed  and  fitted  they  form  a 
complete  circle. 

These  arm  segments  are  then  bolted  between  the  two 
centre  disks,  and  make  a  strong  and  compact  hub. 


INDEX. 


Actual  section  of  ropes,  92 

Adhesion  of  ropes,  168 

Advantage  of  high  rotative  speeds,  157 

Advantages  of  rope-driving,  4,  5 

American  system  of  rope-driving,  24,  25 

Angle  embraced  by  rope,  117 

Angle  of  groove,  164 

Area  of  ropes,  92 

Arras,  number  of,  in  pulleys,  190 

Arms,  shape  of,  199 

Arms,  taper  of,  203 

Atlas  Mills,  12 

Atmospheric  changes,  31 

Automatic  tension-carriage,  28 

Axial  rotation  of  ropes,  174 

Barbour,  James,  built-up  pulley,  205 

Barrus,  G.  H.,  friction  in  mills,  7 

Beeswax  on  ropes,  89 

Belts,  leather,  1,  34,  40 

Blow-holes  in  pulleys,  183 

Bolt  area  for  pulleys,  195 

Braided  rope-joint,  23 

Broadway  Cable  Road,  built-up  pulleys,  217 

Brown,  A.  G.,  friction  loss,  6 

Brush  Electric  Light  and  Power  Co.,  Niagara  Falls,  56 

Built-up  pulleys,  183,  195,  205  to  221 

Catenary  curve,  8,  12 
Cast  grooves,  183 
Centrifugal  force,  45 

Centrifugal  force,  influence  of,  111,  119,  163 
Combe  &  Barbour,  early  rope-drives,  2 

Combe,  Barbour  &  Combe,  basis  for  calculating  horse-power,  99 

223 


224  INDEX. 

Cotton  ropes,  77,  81 

Cotton  fibre,  78 

Cotton  wax,  78 

Continuous- wind  system,  25 

Cone-pulleys  for  ropes,  35 

Coil-friction,  44,  48 

Corliss  engine  and  use  of  jack-shaft,  41 

Coefficient  of  friction,  112 

Cost  of  ropes,  101,  122 

Coulter,  Dr.  S.  M.,  tests  on  inanilla  fibre,  82 

Coupling,  cut-off,  27 

Coupling  for  braided  rope,  23 

Creep  of  ropes,  174 

Cross-section  of  rope,  92 

Cut-off  coupling,  27 

Deflection  of  rope,  131,  170 
Degree  of  twist  in  ropes,  86 
Details  of  rope-pulleys,  205  to  221 
Diameter  of  bolts  for  pulleys,  195 
Diameter  of  pulleys,  103,  161,  177 
Differential  driving,  36,  170 
Differential  pulley,  Walker's,  173 
Double  arms  in  pulleys,  193 
Double  ropes  recommended,  108 
Draw-rods,  71 

Durie,  James,  on  rope-driving,  3 
Dyblie's  rope-tightener,  30 
Dynamo-driving,  38 

Early  use  of  ropes  for  driving,  2,  64 

Effect  of  centrifugal  force,  45 

Effect  of  tar  on  ropes,  91 

Effect  of  wedging  in  groove,  162 

Efficiency  in  any  given  case,  73 

Elasticity  of  ropes,  4,  162 

Elastic  slip  of  ropes,  174 

Engines,  friction  in,  6,  142 

English  rope  system,  4 

Executed  rope  transmissions,  96 

Experiments  on  friction,  141 

Examples  of  large  rope- wheels,  205  to  221 


INDEX.  225 


Fairbanks,  Morse  &  Co.,  split  pulley,  193 

Fibrous  ropes,  75 

Fibre,  cotton,  78 

Fibres  of  manilla,  82 

Flax  ropes,  77 

Fly-wheels,  heavy,  5 

Frictional  grip,  167 

Friction  and  stress  moduli,  118 

Friction-clutch,  33,  42 

Friction,  coil,  44,  48 

Friction,  coefficient  of,  112 

Friction  loss,  141,  158 

Friction  of  engines,  6 

Friction  of  shafting,  6,  68,  145 

Gear  wheels,  1,  5,  8 
Graphite  on  ropes,  89,  91 
Gregg,  multiple  sheaves,  36,  38 
Groove,  angle,  164 
Groove,  shape  of,  186 
Groove,  surface  of,  183 
Groove,  wedging  of  rope  in,  162 
Guide-pulleys,  189 

Harmonic  vibration,  105 

Heavy  liy-wheels,  5 

Hemp  ropes,  2,  77,  90 

Henthorn,  J.  F.,  friction  in  mills,  7 

Hick,  Hargreaves  &  Co.,  built-up  pulley,  208 

Him,  C.  F.,  transmission  of  power,  63 

lloadley  Bros.,  Power-house,  Chicago  City  Ry.  Co.,  53 

Horse-power  of  ropes,  111  to  121 

Hubs  of  pulleys,  191,  203 

Hunt,  C.  \V.,  form  of  splice,  20 

Idlers,  34,  36 

Inclined  transmissions,  139 

Influence  of  belt  pull,  157 

Influence  of  centrifugal  force,  111,  119,  163 

Introduction  of  rope  driving,  2 

Jack-shaft,  use  of ,  for  dynamo-drives,  38,  42 


226  INDEX. 

Jaw  clutch,  42 

Joint  for  braided  ropes,  23 

Kircaldy,  tests  on  ropes,  80 

Lambeth  ropes,  80,  88 

Lanett  Mills,  12 

Large  rope-wheels,  205  to  221 

Laxey,  large  overshot  wheel  at,  71 

Least  diameter  of  pulleys,  103,  161,  179 

Leather  belts,  1 

Length  of  manilla  fibre,  86 

Life  of  ropes,  107 

Limit  of  length  in  shafting,  09 

Linseed-oil  on  ropes,  89 

Link-belt  Co.,  Western  Electric  Co.'s  Plant,  25 

Link-belt  Co.,  Virginia  Hotel  Plant,  42 

Liverpool  Overhead  Railway,  40 

Lockwood  &  Greene,  Lanett  Mills,  12 

Lockwood  &  Greene,  Naumkeag  Mills,  12 

Long-distance  transmission,  62 

Loss  due  to  bending,  159 

Loss  due  to  friction,  141,  158 

Loss  due  to  winder-pulleys,  50 

Losses  in  ropes,  141,  159 

Lubrication  of  ropes,  79,  88 

Manilla  fibre,  82,  86 
Manilla  ropes,  77 

Marlin-spike,  improved  form  of,  23 
Method  of  joining  arms  and  rim,  204 
Miller,  T.  S.,  plant  of  Western  Electric  Co.,  25 
"          "      varying  angle  of  groove,  168 
"          "      use  of  small  ropes,  100 
"          "      use  of  loose  idler,  36 
Multiple  idle  sheaves,  36,  38 
Multiple-rope  system,  4,  17 
Musgrave  &  Sons,  Atlas  Mills,  12 
"     •    "     "    ,  Nevsky  Mills,  16 
"        "      "   ,  rope-pulley,  185 
"         "      "    ,  life  of  cotton  ropes.  107 


IKDEX. 

Naumkeag  Mills,  12 

Nevsky  Mills.  16 

Normal  working  load,  98,  100 

Outdoor  transmission,  50,  66 
Overshot  wheel  at  Laxey,  71 

Pine  tar  as  lubricant,  89 

Pitch  diameter  of  pulley,  181 

Power  absorbed  by  friction  in  shafting,  152 

Power  absorbed  by  ropes  and  gears,  6 

Power  transmitted  by  shafting,  153 

Pulleys,  diameter  of,  38,  103,  161,  177,  181 

Pulley,  tightener,  48 

Pulleys,  supporting,  64 

Pulley,  winder,  47  to  54,  66 

Pulleys,  wood,  56,  164 

"      ,  with  double  arms,  193 

"      ,  with  steel  arms,  185 

"      ,  very  wide,  203 

Rawhide  ropes,  25,  75 
Relative  cost  of  ropes,  122  to  127 
Relative  wear  of  ropes,  122 
Rim  sections,  166,  187 
Ropes,  cotton,  77,  81 

"     ,  cost  of,  101,  122 

' '     ,  early  use  of,  2 

"     ,  elasticity  of,  162 

"     ,  fibrous,  75 
,  flax,  77 

"     ,  hemp,  2,  77,  90 

"     ,  horse-power  of,  121 

"     ,  Lambeth,  80,  88 

"     ,  life  of,  107 

"     ,  lubrication  of,  79 

"     ,  manilla,  77 

"     ,  rawhide,  25,  75 

"     ,  round  leather,  76 

"     ,  shrinkage  of,  31 

' '     ,  speed  of,  1 03 

"     ,  splicing  of,  18 


228  INDEX. 

Ropes,  steel  and  leather,  76 

"     ,  stevedore,  90 

"     ,  strength  of,  79,  91 

"     ,  square  leather,  76 

"     ,  wear  of,  101,  104,  123 

"     ,  weight  of,  109 

"  ,  wire,  75 
Rope  wells,  10 
Rotation  of  ropes,  174 

Sag  of  ropes,  131,  170 
Sections  of  arms,  199 

"         "  rim.  187 
Section  of  rope,  92 
Semicircular  grooves,  164 
Sewall,  M.  W.,  built-up  pulleys,  2tt 
Shafting,  for  long-distance  transmissions,  68 

' '  friction  of,  145 

limit  of  length,  69 
loss  due  to  friction,  68 

"  power  transmitted,  153 

Shafts,  jack,  use  of,  for  dynamo-drives,  38,  42 

"       at  an  acute  angle,  3^ 

' '       at  right  angles,  36 
Shrinkage  of  ropes,  31 
Side  lead  of  ropes,  35 
Size  of  pulley,  38 

"    "  ropes  in  use,  100,  107 
Slack-side  tension,  til 
Small  ropes,  24 
Speed  of  ropes,  103 
Split  hubs,  191 
Splicing  of  ropes,  18 
Stevedore,  transmission  rope,  90 
Stress  in  rim-bolts,  196 

"      "  ropes,  67 
Strength  of  ropes,  79,  91 
Supporting  pulleys,  64 
Surface  of  groove,  183 

Taper  of  arms,  203 


INDEX.  229 


Tallow  on  ropes,  89 
Tar,  effect  of,  on  ropes,  91 
Telodynamic  transmissions,  63 
Temporary  installations,  52 
Tension  carriage,  27 

*'       weight,  28,  32,  140 
"       in  ropes,  111  to  118,  132 
Tests  on  ropes,  80 
Tightener,  29.  30 

"          pulley,  48 
Transmissions  at  an  angle,  34 
outdoor,  50,  66 

"  of  power  to  a  distance,  62 

"  telodynamic,  63 

Turbines,  Victor,  56 
Twist  in  ropes,  86,  88 

Use  of  ropes  with  portable  tools,  62 
Use  of  water-wheels,  56,  66,  71 

Van  Vleck,  built-up  pulley,  214 
Vibration  in  ropes,  105 
Victor  turbine,  56 

Walker,  differential  pulley,  173 
Walker  Mfg.  Co.,  built-up  pulley,  210,  220 
Watertown  tests  on  ropes,  80 
Water-wheels  and  rope-driving,  56,  66 
Wear  of  ropes,  83,  101,  104,  122 
Wear  due  to  side  lead,  35 
Weakening  effect  due  to  twisting,  86 
Webber,  Samuel,  early  use  of  ropes,  2 

"  "         friction  of  shafting,  8 

Wedging  action,  162,  169 
Weight  for  tension-carriage,  139 
Weight  of  ropes,  109 
Wells,  rope,  10 

Western  Electric  Co.'o  plant,  25 
Wetherill  &  Co.,  built-up  pulley,  221 
Wide  pulleys,  203 
Willamette  Mills,  66 
Winder-pulley,  47  to  54,  6.6 


230 


INDEX. 


Wire  ropes,  75 

Wood  pulleys,  56,  164,  186 

Wood-filled  pulley  riin,  186 

Working  strength  of  ropes,  95,  98 

Wound  system  of  rope-driving,  25 

Wren's  "Instrument  for  drawing  up  great  weights,"  45 


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